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Giorgio Mantica  
Center for Non-linear and Complex Systems  
Universit\'a dell'Insubria  
Via Valleggio 11, 22100 Como Italy  
giorgio@uninsubria.it  
 
 Sandro Vaienti  
 Centre de Physique Th\'eorique, Luminy, 
 Marseille and PHYMAT, Universit\'e de Toulon et du Var, France, 
 and F\'ed\'eration de Recherche des Unit\'es de Math\'ematiques 
 de Marseille 
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singular measures, Fourier transform, dimensions, 
orthogonal polynomials, Mellin transform, asymptotic 
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\begin{document}
\title{The Asymptotic Behaviour of the Fourier
Transforms of Orthogonal Polynomials I: Mellin transform
techniques}
\author{Giorgio Mantica and Sandro Vaienti}
%
\date{ }
\maketitle
%
\begin{abstract}
The Fourier transforms of orthogonal polynomials with respect to
their own orthogonality measure defines the family of
Fourier-Bessel functions. We study the asymptotic behaviour of
these functions, and of their products, for large real values of
the argument. By employing a Mellin analysis we construct a
general framework to exhibit the relation of the asymptotic decay
laws to certain dimensions of the orthogonality measure, that are
defined via the divergence abscissas of suitable integrals. The
unifying role of Mellin transform(s) techniques in deriving
classical and new results is underlined.

\end{abstract}

{\em 2000 Math. Subj. Class: 42C05, 33E20, 28A80, 30E15, 30E20}

{\em Keywords: singular measures, Fourier transform, dimensions,
orthogonal polynomials}


\section{Introduction}\label{intro}

This is the first paper of a series on the asymptotic behaviour of
the Fourier transforms of the orthogonal polynomials of a measure.
Let $\mu$ be a Borel measure that we suppose to be normalized, and
for which the {\em moment problem} is determined \cite{achie}:
that is, all moments $\mu_n:=\int d \mu(s)~s^n$, $n\in
{\bf N}$ exist, and uniquely identify the measure. This
includes measures with unbounded support, as well as compactly
supported multi-fractal measures, whose properties are
particularly interesting. We then consider the set of associated
{\it orthogonal polynomials}
$\{p_0(\mu;s),p_1(\mu;s), \ldots \}$,
 \beq
 (p_n, p_m) :=\int d\mu(s)~p_n(\mu;s) p_m(\mu;s)=\delta_{nm},
  \nuq{opol1}
where $\delta_{nm}$ is the Kronecker delta, and $(\cdot,\cdot)$ is
the scalar product in $L^2({\bf R},d \mu)$.
Let us now define:
 \begin{defi}
 The generalized Fourier-Bessel
Functions (F-B. functions for short) ${\cal J}_n(\mu;t)$ are the
Fourier transforms of the orthogonal polynomials $p_n(\mu;s)$ with
respect to $\mu$ itself:
 \beq
   {\cal J} _n(\mu;t) :=
     \int d\mu(s) ~p_n(\mu;s)~e^{-its}.
 \nuq{geb1}
\end{defi}
%
This  paper is concerned with the investigation of the long-time
behaviour of the temporal Cesaro averages of the generalized
F-B.functions,
 \beq
  \bar{\cal J}_n(\mu;t) :=  {\cal C}(J_n;t) := \frac{1}{2t} \int_{-t}^t
{\cal J}_n(\mu;t') \; dt' .
 \nuq{tav}
Here and in the following, ${\cal C}(f;t)$ indicates the symmetric
Cesaro average of a function $f$. This can be rewritten as
  \beq
   \bar{\cal J}_n(\mu;t)  = \int \Phi(t,s)
    p_n(\mu;s) \; d\mu(s) ,
 \nuq{cru2}
where $\Phi(t,s) = \sin (ts) / ts = \mbox{sinc} (ts)$. Eq.
(\ref{cru2}) also makes it evident that $\bar{\cal J}_n(\mu;t)$ is
a real quantity. Comparison with eq. (\ref{geb1}) shows that the
same notation can be employed for instantaneous values, letting
$\Phi(t,s) = e^{-its}$. The techniques presented here can treat
equally well to this case, although with different results
when $\mu$ is singular continuous. In fact, motivation for this
paper is the study of this class of measures: here, recourse to
Cesaro averaging is dictated by the presence of intermittent
oscillations of the {\em F-B.} functions \cite{physd}.


At the same time, we shall study the Cesaro averages of products
of two {\em F-B.} functions:
 \begin{defi}
The quadratic {\em amplitudes} $A_{nm}(\mu;t)$ are defined as:
%
\begin{equation}
 A_{nm}(\mu;t) :=  {\cal C}(J_n J^*_m;t)  := {1 \over 2t} \int_{-t}^{t}
 {\cal J}_n(\mu;t') {\cal J}^*_m(\mu;t')
 \; dt' . \;
 \label{czz}
\end{equation}
\end{defi}
%
This problem, in the case $n=m=0$ is classical in the literature,
and a variety of techniques for its solution have been proposed,
of those we present a brief review in the next section.

In this paper, we aim at presenting a class of techniques based on
the Mellin transform that permits to recover the existing results
into a global theory, and to obtain new ones. For this, we shall
consider the Mellin transform of the Cesaro averages $\bar{\cal
J}_n(\mu;t)$ and $A_{nm}(\mu;t)$, that we shall call $M_n(\mu;z)$
and $M_{nm}(\mu;z)$, respectively, letting the number of
subscripts discriminate between the two cases. We adopt the
following convention for  the definition of the Mellin transform
of a function $f$:
  \beq
   {\cal M}(f;z) := \int_0^{\infty} dt ~f(t) ~t^{z-1}.
  \nuq{mel1}
The precise meaning of this integral will be defined in the
following, as convenience will demand, either as a Lebesgue
integral, or as an improper Riemann integral.

Let now $f$ be positive: in our case, the diagonal amplitudes
$J_n J^*_n$ are such. Then, it is well known that ${\cal
M}(f;z)$ is analytic in a strip $\zeta_0 < \Re z < \zeta_\infty$.
This domain of analyticity of the Mellin transform is indicative
of the short and long-time behaviour of $f(t)$. Since we are
mainly interested in the latter, we shall investigate the {\em
divergence abscissa} of the Mellin transform, $\zeta_\infty$ . It
is an easy exercise to show that $\zeta_\infty$ interpolates the
upper and lower limits of $- \log f(t) / \log (t)$, for $t
\rightarrow \infty$. Indeed, in a number of cases, equality with
the upper limit follows as a consequence of a classical theorem.
In addition, complex analysis will be employed to treat the case
of the non necessarily positive, or even real, functions
$\bar{\cal J}_n(\mu;t)$ and $A_{nm}(\mu;t)$, with $n \neq m$.

In this analysis, a key role will be played by potential theoretic
quantities. First, we shall consider
\bde
 The generalized
electrostatic potential ${\cal G}(\mu;s,z)$ of the measure $\mu$
at the point $s$ is
 \beq
  {\cal G}(\mu;s,z) = \int \; d\mu(r) \frac{1}{|r-s|^{z}}.
 \nuq{melc2x}
\ede Integrating the potential with respect to $\mu$ leads to:
\bde The generalized electrostatic energy ${\cal E}(\mu;z)$ of the
measure $\mu$ is:
\begin{equation}
  {\cal E}(\mu;z) :=   \dbleint d\mu(r) d\mu(s) {1\over
  |r-s|^{z}}.
\label{elec}
\end{equation}
\ede
 It is immediate to see that eqs. (\ref{melc2x}) and (\ref{elec})
define analytic functions in a half plane. Their {\em divergence
abscissas} can be used to define two important quantities:
\begin{defi}
The electrostatic local dimension $d(\mu;s)$ of a measure $\mu$ at
the point $s$ is the divergence abscissa of the generalized
electrostatic potential ${\cal G}(\mu;s,z)$:
 \[
  d(\mu;s) := \sup \{x
\in {\bf R} \; \mbox{s.t.} \;  {\cal G}(\mu;s,x) < \infty \} .
\]
\label{def0}
\end{defi}
%
\begin{defi}
The electrostatic correlation dimension $D_2(\mu)$ of a measure
$\mu$ is the divergence abscissa of the generalized electrostatic
energy ${\cal E}(\mu;z)$:
 \[
  D_2(\mu) := \sup \{x \in {\bf R} \;
\mbox{s.t.} \; {\cal E}(\mu;x) < \infty \} . \]
 \label{def1}
\end{defi}
We observe that these electrostatic dimensions coincide with the
corresponding {\em lower} dimensions, as defined by the
thermodynamical formalism, thanks to \cite{yorke}, \cite{hunt}.
Indeed, these results are a consequence of a quite general result
on the Mellin transform of Stieltjes measures that is implicit in
the literature, and that we shall spell out and prove in the
following.

We will show in this paper that the electrostatic dimensions are
related to the rightmost divergence abscissas of $M_{n}(\mu;z)$
and $M_{nm}(\mu;z)$; therefore, they provide the asymptotic behaviour
of $\bar{\cal J}_n(\mu;t)$ and $A_{nm}(\mu;t)$  for large $t$,
for any measure $\mu$.
Proper analytical tools will then be
utilized and developed to provide a precise meaning to the
asymptotic relation. In addition, relations between the two
asymptotic behaviours will be brought to light.

In a companion paper, the theory will be made explicit for the
balanced invariant measure of an Iterated Function System, and the
results will be applied to quantum mechanics.

This paper is organized as follows: in the next section we shall
attempt a critical discussion of the results that have already
appeared in the literature on particular subcases of our problem,
or that have dealt with a restricted set of situations. In the
remainder of the paper we hope to convince the reader that the
Mellin approach achieves the widest generality. In section
\ref{uplosec} we prove a theorem of which the results of
\cite{yorke}, \cite{hunt} mentioned above can be seen as a
particular case. A variant of this theorem is applied in Sect.
\ref{lontim} to re-derive a classical result on the decay of the
Fourier transform of a measure.

We then start the analysis of the Mellin transform of the averaged
F-B. functions. In sect. \ref{local} we introduce suitable
asymptotic exponents and we prove a general decay theorem that
relates these latter to the Cesaro averages of F-B. functions.
Variants of this theorem will be proven with different techniques
(still based on the Mellin transform) in the successive sections.
In Sect. \ref{ana3} we put in relation the local properties of the
measure $\mu$ at zero and the analyticity structure of the Mellin
transform. This information is then utilized in Sect. \ref{invmt}
to set up an inverse transform technique. This leads to the proof
of a power-law bound on the asymptotic decay of the averaged F-B.
functions, in a restricted range of exponents. In Sect.
\ref{elle2} the inversion theory, in a weak form originally due to
Makarov \cite{maka}, is applied in a larger set of exponents.
 Next, in Sect. \ref{strong},
an elementary application of integration by parts in a Riemann
integral, combined with a computation of the Mellin transform of
{\em non}--averaged F.B. functions, permits to obtain the
asymptotic decay in strong form over a larger set of exponents.

Starting from Sect. \ref{analy} we turn our attention to the
Cesaro averaged quadratic amplitudes $A_{nm}(\mu;t)$, much in 
line with the case of the previous sections. We first put in
relation the analyticity structure of their Mellin transform with
the dimensional properties of the measure $\mu$ and we prove a
general decay theorem. These properties are then utilised in sect.
\ref{invqua} to set up the Mellin inversion procedure. Positivity
properties of the {\em diagonal} quadratic amplitudes (that is,
the amplitudes of physical significance in quantum mechanics) are
employed in sect. \ref{asydpq} to prove the asymptotic decay of in
strong form, for the widest set of exponents. The same strong
result is obtained in Sect. \ref{asycross} for the non-diagonal
amplitudes, still following the techniques developed for linear
quantities.


A further object is analyzed in Sect. \ref{asymm}: the asymptotic
behaviour of the Mellin transform of the F-B. functions when the
argument goes to infinity in the vertical strip of analyticity. We
show that the techniques  of the previous sections can be applied
to these quantities, by introducing a class of measures,
$\nu_x(\mu)$, constructed upon the original measure $\mu$, and the
abscissa $x$ in the analiticity strip. Under the restrictive
hypothesis of boundedness of the support of $\mu$, the relations
between $D_2(\mu)$ and $D_2(\nu_x(\mu))$ are investigated. In
sect. \ref{ciccio} we show how this analysis can be employed to
extend the $L_1$ property of the Mellin transform.

Finally, in Sect. \ref{loarb} we show how to adapt the analysis
to the local behaviour of the measure $\mu$ around any arbitrary
point. We derive a lower bound on the local dimensions at all
points that is
equal to one half of the correlation dimension. The conclusions, Sect.
\ref{conc}, briefly discuss the relevance of the Mellin analysis
of spectral and dynamical properties like those encountered in
this paper. An elementary appendix exemplifies the results of this
paper on a simple sequence of measures.



\section{Discussion of Previous Results and of the Mellin Transform Approach}
\label{secmel}

It is now important to review previous results that apply to our
problem. We shall not proceed in chronological order, but rather
we shall move from more specific to more general results. Of
course, we cannot claim completeness, and we apologize for
involuntary omissions. First of all, since $p_0(\mu;x)$ is a
constant, the case $n=m=0$ leads to the Fourier transform of
$\mu$. This case has obviously received a lot of attention.

The first result is Strichartz theorem \cite{str0}: it underlines
the importance of the continuity properties of the measure $\mu$.
A measure $\mu$ is called locally uniformly $\alpha$-dimensional
if there exist a constant $C$ such that the measure  of the ball
of radius $r$ centered at $s$, $\mu(B_r(s))$, is bounded by $C
r^\alpha$ uniformly in $s$. In a sense, $\alpha$ is a sort of
lower bound to the local dimensions of $\mu$. Under these
conditions, Strichartz theorem (adapted to the $n,m$ case)
predicts that
 \[
     \limsup_{t \rightarrow \infty}  \; t^{\alpha} \; A_{nm}(\mu;t)  \leq C',
  \]
with $C'$ another constant. A reverse inequality, this time with
the {\em liminf}, can be obtained when the support of the measure
is a {\em quasi-regular} set \cite{str0,lau1}. Therefore, when
these conditions are met, Strichartz's analysis provides us with
the leading asymptotics of the amplitudes: $A_{nm}(\mu;t) \sim
t^{-\alpha}$.

It is a matter of fact that most interesting measures are {\em
not} uniformly $\alpha$-dimensional measures, or with
quasi-regular support; typically, {\em
multi-fractal} measures do not possess these properties.
Among these, there exists a family for
which the problem has been solved to a large extent, the class of
Linear Iterated Functions Systems (L.I.F.S.), that we discuss in a
second paper. Various techniques have been used to tackle this
class, all of them relying on the  self-similarity properties of
the measure \cite{l1l2,l2,lw1,str2}. The exponent  $\alpha$ of the
asymptotic decay law is then determined by an implicit equation.
It turns out, obviously, that this value coincides with the
correlation dimension of the L.I.F.S. measure. The same result for
disconnected L.I.F.S. has also been obtained following a different
route, with the aid of the Mellin transform \cite{turca}
\cite{maka},\cite{physd}.

The approach via the Mellin transform has the advantage of
rendering the identification of the decay exponent
transparent. In addition, it does
not require any open set condition---except, of course, if one
wants to compute explicitly such dimension. One of the
goals of this paper is to show that the Mellin transform is fully
general, and applies to any probability measure $\mu$, for which
the moment problem is determined. Indeed, were this {\em not} the
case, the results for the $n=0$ case would hold unchanged.



The Mellin approach is implicitly contained in some estimates on
the correlation dimension in Falconer's book \cite{falco}, but it
was firstly employed in the present context by Bessis {\em et al.}
in \cite{turca}, that offers the earliest explicit
proof, to our knowledge,
of the relation between the asymptotic behaviour of the Fourier
transform of a measure and its correlation dimension, defined as
in section \ref{intro}. Ref. \cite{turca} also contains the case
of linear I.F.S., that was further analyzed in \cite{turorl}.
Since the main focus of \cite{turca} was on correlation integrals,
the result may have passed unnoticed. As a matter of fact, {\em
three} problems are intimately related, and lead to the
generalized energy integral ${\cal E}(\mu;z)$ of eq. (\ref{elec}):
the asymptotic behavior of the Fourier transform of a measure,
that of correlation integrals, and that of quantum amplitudes, for
which Ketzmerick {\em et al.} \cite{ketze} rediscovered the role
of the correlation dimension by using formal manipulations.  These
latter were further made rigorous and extended by the use of
functional analysis in \cite{jmbarb}, and by wavelet techniques in
\cite{guer2}, to show that $ \limsup(\inf)_{t \rightarrow \infty}$
$ \log A_{00}(t)/ \log t$ coincide with the upper and lower
correlation dimension of the measure $\mu$. In the quantum
mechanical context, further results were established in
\cite{last}, that we shall mention later, since they also use
Mellin-like techniques. Also concerned with the relation between
the continuity properties of the measure, and its Fourier
transform is the work \cite{hof}.

Linear quantities, leading to the local rather than the
correlation dimension, have been studied in the wavelet theory
\cite{wavsa,wavsa2}, and it the Green function analysis of
measures of quantum mechanical origin \cite{belli2}.

 In \cite{physd}  \cite{france} the Mellin
technique was applied not only to the quantities discussed in this
paper, but also to a more cumbersome family of quantities, still
constructed out of the quantum mechanical amplitudes: the sums
$\nu_\alpha(t)$ $ := \sum_n n^\alpha |{\cal J} _n(\mu;t)|^2$.
These are the moments of the quantum distribution over a discrete
lattice, and the asymptotic behaviour of the sum of the series is
different from that of its individual terms.

The results we present here widen the scope of these
investigations in many respects: firstly, our results apply to the
case of a general measure, as opposed to the specific case of
L.I.F.S. measures. This specific class being of particular
importance, it will be further analyzed in a companion paper.
Moreover, we shall extend the analysis from the Fourier transform
of the measure alone, to that of the related orthogonal
polynomials, for all values of the index. In addition, we shall
also investigate the non-positive quantities arising from
averaging the F-B. functions themselves, a theme that has received
less attention than the quadratic case. Also new is the
analysis of the asymptotic behaviour of the Mellin transform
itself, in the strip of analyticity, that leads to interesting
speculations on a different family of measures, that we shall
only briefly consider in this paper. Finally, we believe that
collecting old and new results under the unifying theme
of the Mellin transform(s) techniques is a valuable endeavour,
that might bring benefit in different areas of research.


\section{Electrostatic, and generalized dimensions}
\label{uplosec}

The relation of the Mellin dimensions defined in Sect.
\ref{intro}, and the more conventional generalized dimensions is
well known. This can be seen as a particular case of a general
theorem on Mellin (and Laplace) transform that is presented in
this section.
%
 \bde
 \label{def5.2}
 The upper (lower) local dimensions
$\gamma_\pm (\mu;s)$ of a measure $\mu$ at the point $s \in {\bf
R}$, are defined by
 \beq
  \gamma_\pm (\mu;s) = \lim \sup (\inf) _{\epsilon \rightarrow 0}
             \frac{\log \mu ((s-\epsilon,s+\epsilon)) }{\log \epsilon}.
\nuq{locdim}
 \ede

\bde
 \label{defcord}
 The upper (lower) correlation  dimensions
$D_2^\pm (\mu)$ of a measure $\mu$, are defined  by
 \beq
  D_2^\pm (\mu) = \lim \sup (\inf) _{\epsilon \rightarrow 0}
             \frac{\log
     \int d\mu(s)
              \mu ([s-\epsilon,s+\epsilon]) }{\log \epsilon}.
\nuq{cordim}
 \ede

\bth [\cite{yorke}, \cite{hunt}] .
  \label{hunt,yorke}
  The electrostatic dimensions $d(\mu;s)$ and $D_2(\mu)$ coincide with the
   lower dimensions: $\gamma_- (\mu;s) = d(\mu;s)$, $D_2^- (\mu) =
   D_2(\mu)$.
  \eth

As a matter of fact, Theorem \ref{hunt,yorke} is a consequence of
a general theorem on Mellin (and Laplace) transforms that can be
found more or less explicitly in the literature. A sketch of its
proof will be helpful:

\bth \label{teosmall} Let $m(u)$ a Stieltjes measure on $[0,1]$,
such that $m(0)=0$, $m(1) < \infty$, and let $M(m;z)$ be the
Mellin-Stieltjes transform of $m$:
 $
  M(m;z) := \int_0^1 u^{-z} dm(u).
  $
 Let $d(m)$ be
the divergence abscissa of $M(m;z)$. Clearly, $d(m) \geq 0$.
Moreover,
 \beq
 l := \lim \inf_{\epsilon \rightarrow 0}
             \frac{\log m(\epsilon) }{\log \epsilon} = d(m).
\nuq{teosm1} \eth
%
{\em Proof:} Notice the a different definition adopted for the
definition of the MT of a measure on $[0,1]$, when compared to the
usual eq. (\ref{mel1}). Take $x \in {\bf R_+}$, $x<d(m)$. Then,
$M(m;x)$ is finite. This implies the chain of inequalities $
  C \geq  \int_0^\veps u^{-x} dm(u) \geq \veps^{-x} \int_0^\veps
  dm(u)= \veps^{-x} m(\veps)
  $
for all $\veps$. As a consequence, $\frac{\log m(\epsilon) }{\log
\epsilon} \geq x + \frac{\log C}{\log \epsilon}$, that implies $l
\geq d(m)$. Let us now show the reverse inequality. Because of the
definition of lower limit, for any $\eta > 0$, there exist
$\veps_\eta$ such that, for any $\veps < \veps_\eta < 1$,
$m(\veps) \leq  \veps^{l - \eta}$. By a standard Laplace-type
estimate we can write:
 \[
   M(m;x) =
\! \!
   \int_0^{\veps_\eta} \! \! \! \! \! dm(u) u^{-x}
              + \int_{\veps_\eta}^1  \! \! \! \! \! dm(u) u^{-x}
            \leq
 u^{-x} m(u) |_{0^+}^{\veps_\eta}
  \! \!
   + x \int_0^{\veps_\eta} \! \! \! \! \! u^{-x-1} m(u) du
   + C \veps_\eta^{-x}.
   \]
The last term at r.h.s. is obviously finite. Let us consider the
first term. The contribution at $\veps_\eta$ is also finite.
Moreover, $u^{-x} m(u) \leq u^{-x+l-\eta}$, so that $\lim_{u
\rightarrow 0^+} u^{-x} m(u)$ is null when $x < l-\eta$. For the
same reason, under this condition the remaining integral term is
also finite, and so is $M(m;x)$. This holds for all $\eta$, and
therefore the divergence abscissa is larger than, or equal to,
$l$. \qed \\
%
{\em Proof of} Thm. \ref{hunt,yorke} follows  from Thm.
\ref{teosmall}. For the local dimension, the Stieltjes measure is
$m(u) := \mu([s-u,s+u])$, and as such it is closely related to
Theorem 6.4 in \cite{falco} -- albeit we use a slightly different
technique. For the correlation dimension, one lets $m(u)$ be the
correlation measure $\Omega(\mu;u)$ defined via
 \beq
  \Omega(\mu;u) := \int \! \! \! \! \! \int_{|s-r| \leq u} d \mu(s) d
  \mu(r),
  \nuq{cordint}
  and the result follows. \qed
\bre {\rm
Formula (\ref{cordint})
above shows that the correlation dimension can be equivalently
considered as the local dimension, at the point zero, of the
correlation measure associated with $\mu$. Therefore, the results
that we shall obtain for the local dimensions will immediately
extend {\em mutatis mutandis} to the correlation dimension.
}\ere

\section{Long time limits: Laplace-like results}
\label{lontim}

The well known asymptotic behaviour of the Fourier transform of a
measure can be also seen as a particular case of the general Thm.
\ref{teosmall}. For this, we need a close analogue of this latter:

\bth \label{teolarge} Let $m(u)$ a Stieltjes measure on
$[1,\infty)$, and let the Mellin-Stieltjes transform of $m$,
$M(m;z)$, be defined here as:
 \[
  M(m;z) = \int_1^\infty u^{z-1} dm(u);
  \]
let $d(m)$ be its divergence abscissa. Put $\alpha(m) := \min
\{d(m),1\}$. Let  ${C}(m;r)$, the Cesaro average, be defined as

 \[
  { C}(m;r) := \frac{1}{r}\int_1^r d m(u),
  \]
and finally let $L := \limsup_{r \rightarrow \infty} \log {
C}(m;r) / \log r$. Then,
 \beq
 -d(m) \leq L \leq - \alpha(m).
 \nuq{inelarg1}
 \eth
%
{\em Proof:} We sketch only the differences with the analogue
Theorem \ref{teosmall}. The first part of the proof differs only
in the fact that, in order to get the inequality $r^x { C}(m;r)
\leq C$ one must require that $x < d(m)$ and, in addition, $x \leq
1$. In so doing, the second inequality in (\ref{inelarg1})
follows. The second part of the proof is basically unchanged, with
the sole substitution of liminf with limsup, and leads to the
first inequality in the thesis.
 \bre {\rm
It is easy to see that the thesis can be equivalently stated in
terms of the usual Cesaro average ${\cal C}(f;t)$, when the
integral of $dm(u) = f(u) du$ between zero and one can be
controlled.
 } \ere

This theorem highlights the basicality of the relation between the
asymptotic of the Fourier transform and the correlation dimension:
 \bth
\label{teolarge2} Let $dm(t) = |{\cal J}_0(\mu;t)|^2 dt$, so that
${ C}(m;t)$ is the  Cesaro average of the square of the Fourier
transform of the measure $\mu$. Then,
 \[ \limsup_{t \rightarrow
\infty} \log { C}(m;t) / \log t = D_2(\mu).
 \]
 \eth
%
{\em Proof:} We need to compute the Mellin transform
 \[
M(m;z) = \int_1^\infty t^{z-1} dm(t) = \lim_{T \rightarrow \infty}
 \int_1^T \! dt \;\; t^{z-1}
 \dbleint d \mu(r) d \mu(s)
   e^{-it(r-s)}.
 \]
This can be also written as
 \[
\lim_{T \rightarrow \infty} \dbleint d \mu(r) d \mu(s)
\frac{1}{|r-s|^z} \int_{|r-s|}^{|r-s|T} u^{z-1} e^{-iu} du,
 \]
thanks to a change of variable, and requiring that $\Re z <
D_2(\mu)$, so that the integrand of the triple integral is
summable, to apply Fubini theorem. Now, observe that when $0 \leq
\Re z \leq 1$ the last integral in the above expression is a
bounded function of all three variables $r,s$ and $T$, so that
$D_2(\mu)$ is the divergence abscissa of the Mellin transform
$M(z)$. To conclude the proof, apply Thm. \ref{teolarge}. \qed

\bre \label{last} {\rm This theorem is fully equivalent to the
analysis of Last, Lemma 5.2 of \cite{last}, and the remark just
following. Scrutiny of these proofs shows their equivalence, via
Thm. \ref{teolarge}, to the proofs of Thm. \ref{hunt,yorke} in
\cite{yorke}, \cite{hunt}. Our proof of Thm. \ref{hunt,yorke}
differs in one of the inequalities, and is closer to the standard
usage in Laplace transforms.} \ere

\section{Local Properties of the Measure and Asymptotic Decay
of F-B. functions.} \label{local}

We now start the analysis of the asymptotic behaviour of the
Cesaro averages $\bar{\cal J}_n(\mu;t)$. We shall find that they
are related to the local properties of the measure $\mu$ at zero.
This will entail notational simplification with respect to the
previous section, by dropping the explicit reference to the point
zero. A movable local analysis will be reintroduced starting from
Sect. \ref{loarb}.

In addition,
notice that throughout this paper we shall write $z=x+iy$, with
$x$ and $y$ real.
 We
start by re-writing the function $M_n(\mu;z)$, eq. (\ref{mel1}),
in a convenient way. Firstly, we observe that the
integral
 \beq
I_n (\mu;x) =  \int d \mu(s) \int_0^\infty
    dt \; |t^{x-1} p_n(\mu;s) \frac{\sin ts}{ts} |
\nuq{fub1}
 is convergent, when $x$ belongs to a suitable interval.
In fact, we can split the inner integral in two parts, obtaining
  \beq
   \ba{ll} I_n(\mu;x) & \leq   \int d
\mu(s) |p_n(\mu;s)| [
  \int_0^{\frac{\pi}{2|s|}} dt \; t^{x-1} +
  \int_{\frac{\pi}{2|s|}}^\infty  dt \;
  t^{x-1} \frac{1}{t|s|} ] \\
  & =
  [  \frac{1}{x} (\frac{\pi}{2})^{x}
  + \frac{1}{1-x} (\frac{\pi}{2})^{x-1}]
  \int d \mu(s) |p_n(\mu;s)| |s|^{-x} ,\\
\ea
 \nuq{fub3}
 where we are forced to restrict $x$ to
$(0,1)$, being the r.h.s. infinite otherwise. Therefore, if
$0<x<1$ and the integral
 \beq
  {\cal G}_n(\mu;x) = \int \; d\mu(s) \frac{|p_n(\mu;s)|}{|s|^{x}},
 \nuq{melc2xb}
 is convergent, so is $I_n(\mu;x)$.  The integral ${\cal
G}_0(\mu;x)$ has been introduced previously: it is the generalized
electrostatic potential at the point zero. It divergence abscissa,
$d_0(\mu)$, defines the local dimension \cite{belli}. This latter
is certainly larger than, or equal to zero, since $\mu$ is a
probability measure, and $p_n(\mu;s)$ are its orthogonal
polynomials. We now need to modify the definitions to let these
latter play a role:
 \bde
  \label{def5.1a}
%
The  electrostatic local dimensions of $\mu$ at zero, $d_n(\mu)$,
are the divergence abscissas of the integrals ${\cal G}_n(\mu;z)$,
 \beq
   d_n(\mu) := \sup \{ x \in {\bf R} \;  \mbox{s.t.} \;\; {\cal G}_n(\mu;x) < \infty \}.
 \nuq{tildal}
 \ede
%
\bre {\rm Because of Def. \ref{def0}, this is equivalent to say
that $d_n(\mu)$ is the local dimension, at zero, of the measure
$\nu = |p_n| \mu$: $d_n(\mu) = d(|p_n| \mu)$. } \ere
%
  \bde \label{asyespo}
 The asymptotic exponents $\alpha_n(\mu)$  are the
divergence abscissas of the integrals $I_n(\mu;z)$:
 \[
  \alpha_n (\mu):= \sup \{  x \in {\bf R}  \; \mbox{  s.t. }
 I_n(\mu;x) < \infty \}.
 \]
 \ede
%
 We are now in the condition to prove  that
 \bpr
 \label{lem5.1}
 If $p_n(0) \neq
0$, the dimension $d_n(\mu)$ is equal to $d_0(\mu)$. If
$p_n(0) = 0$, the dimension $d_n(\mu)$ is equal to
$d_0(\mu)+1$.
 Moreover,
  $
\alpha_n (\mu)=  \min \{ d_n(\mu),1 \}.
   $
  \epr
%
{\em Proof.} Let $x \geq 0$, $a>0$.
 \beq
 {\cal G}_n(\mu;x)  = \int_{|s| < a} \; d\mu(s) \frac{|p_n(\mu;s)|}{  |s|^{x}}
  + \int_{|s| \geq a} \; d\mu(s) \frac{|p_n(\mu;s)|}{  |s|^{x}}
\nuq{ia1}
 The second integral defines an analytic function. In
fact, it can be bounded by
 \[
 \frac{1}{|a|^{x}} \int  d\mu(s)  |p_n(\mu;s)|  \leq
 \frac{1}{|a|^{x}}(\int  d\mu(s)
 |p_n(\mu;s)|^2)^{1/2} = \frac{1}{|a|^{x}} ,
 \]
because $\mu$ is a probability measure, and orthogonal polynomials
are normalized.
%
  Consider now the first integral in (\ref{ia1}). If $p_n(0) \neq 0$,
choose $a$ such that on  $[-a,a]$ $p_n(\mu;s)$ is strictly
different from zero. Let $m=\min \{|p_n(\mu;s)|\}$, $M=\max
\{|p_n(\mu;s)|\}$. We have
\[
  m  \int_{|s| < a} \; d\mu(s)  |s|^{-x} \leq
  \int_{|s| < a} \; d\mu(s) |p_n(\mu;s)| |s|^{-x}  \leq
  M   \int_{|s| < a} \; d\mu(s)  |s|^{-x},
\]
and the first part of the thesis follows. When $p_n(0)=0$, recall
that zeros of orthogonal polynomials are always simple, so that:
$p_n(\mu;s)= s q_{n-1}(s)$, with $q_{n-1}(0) \neq 0$, and apply
the same reasoning.

The inequality $\alpha_n (\mu)\geq \min \{ d_n(\mu),1 \}$ follows
upon consideration of eq. (\ref{fub3}). The reverse inequality can
be obtained from
 \[
   I_n(\mu;x)  \geq  \int d
\mu(s) |p_n(\mu;s)|
  \int_{\frac{\pi}{2|s|}}^\infty  dt \;
  t^{x-1} | \sinc (t|s|) | = \]
  \beq =
\int d \mu(s) \frac{|p_n(\mu;s)|}{|s|^{x}}
 \int_{\frac{\pi}{2}}^\infty du \;
   u^{x-2} | \sin (u) |,
    \nuq{fub3c}
that shows that if one of the integrals at r.h.s. is divergent, so
is $I_n(\mu;x)$.
   \qed
%
\bre {\rm The first case considered in the proposition above,
$p_n(0) \neq 0$, is obviously typical. On the other hand,
$p_n(0)=0$ happens for instance when the measure is symmetric with
respect to zero, and $n$ is odd, $p_n(-x) = - p_n(x)$: the
divergence abscissa $d_n(\mu)$ is then infinite. But in this case
the Cesaro average we are studying is null, and our analysis is
empty. A remedy is then to employ a one-sided Cesaro average,
between $0$ and $t$. } \ere
%
The asymptotic exponents $\alpha_n(\mu)$ deserve their name in
virtue of the following fundamental Theorem.
%
 \bth
 \label{thfinal}
 Let $\alpha_n(\mu)$ be the divergence abscissa in Def. \ref{asyespo}.
 For all $x$ such that $x <  \alpha_n(\mu)$, when $t \rightarrow \infty$, one has
$
\bar{\cal J}_n(\mu;t) = o (t^{-x})
$
 \eth
{\em Proof.} Because of equation (\ref{cru2}), one writes
 \[
  t^x \bar{\cal J}_n(\mu;t) = \int d\mu(s) t^x \sinc (ts) p_n(\mu;s) =
\int \frac{d\mu(s)}{|s|^x} |st|^x \sinc (|ts|) p_n(\mu;s).
  \]
Let $x \in [0,1)$. Then the function $u \rightarrow u^x \sinc (u)$
is bounded by a constant on $\bf R$. Therefore, $| |st|^x \sinc
(|ts|) p_n(\mu;s)| \leq C |p_n(\mu;s)|$.  Since $x <
\alpha_n(\mu)$, this latter function is integrable with respect to
the measure ${d\mu(s)}/{|s|^x}$. The dominated convergence theorem
permits then to take the limit for infinite $t$ inside the
integral sign. \qed


\section{Analytic Representation of the MT} \label{ana3}

In this section we investigate the analiticity properties of the
Mellin transform $M_n(\mu;z)$.  The considerations of Sect.
\ref{local} can be regarded as preparatory work to the following:
when $x = \Re(z) < \alpha_n(\mu)$, it is legitimate to exchange
the order of the integrals defining $M_n(\mu;z)$, to obtain:
 \bpr
 \label{lem5.2}
% \\ {\bf Lemma 5.2} {\em
The integral representation
\beq
 M_n(\mu;z) = \int_0^\infty
    dt \; t^{z-1}
  \int d \mu(s) p_n(\mu;s) \frac{\sin ts}{ts}
  \nuq{meldf4}
 defines an analytic function in
the domain $0 < \Re (z)  < \alpha_n(\mu)$. This function can be
also expressed as
 \beq
  M_n(\mu;z) = H(z) \; G_n(\mu;z),
  \nuq{fub4}
  with $H(z)$ the analytic continuation of the Mellin transform
of $\sinc(\cdot)$,
  \beq
   H(z) =
    \Gamma(z-1) \sin [\frac{\pi}{2} (z-1)],
 \nuq{melc1}
  and
  \beq
 G_n(\mu;z) = \int \; d\mu(s) \frac{ p_n(\mu;s) }{  |s|^{z}}.
 \nuq{melc2b2}
 \epr
{\em Proof:}
 Since
 \[
 \int_0^\infty
    dt \;  | t^{z-1}
  \int d \mu(s)  p_n(\mu;s)  \frac{\sin ts}{ts}|
  \leq    I_n(\mu;x) <
  \infty,
 \]
%
the first part of the thesis follows. Moreover, Fubini theorem
applies: changing the order of integration in eq. (\ref{meldf4})
then provides the formulae (\ref{fub4}),
(\ref{melc1}),(\ref{melc2b2}). \qed
%
\bre {\rm
The function $H(z)$  is purely kinematical, and does not depend on
the measure $\mu$. It is a meromorphic function, with simple poles
at $-2 k$, $k=0,1,\ldots$, of residuals $\rho_k = (-1)^k$. The
pole at zero is related to the $O(1)$  behavior of $\sin(t)/t$ for
small $t$. The second term, $G_n(\mu;z)$, explicitly calls in
cause the short scale properties of the measure $\mu$ at zero.
Because of this term, we see that $M_n(\mu;z)$ is convergent, for
$\Re(z) \in [0,1)$, if and only if $G_n(\mu;z)$ is such.
Furthermore, if $d_n(\mu)$ is larger than one, $M_n(\mu;z)$ can be
analytically continued for $\Re (z) > 1$ via eq. (\ref{fub4})
until the first singularity of ${\cal G}_n(\mu;z)$, $d_n(\mu)$. Of
course, this analytical continuation does not imply that the
original Mellin transform integral is convergent in Lebesgue
sense. In Sect. \ref{strong} we shall employ Riemann convergence
to extend the domain of convergence.
} \ere



\section{Inversion of MT and Asymptotic Properties}
\label{invmt}

 We can now return  to the analysis of the asymptotic
behaviour of the integral (\ref{cru2}). The poles of $H(z)$,
possessing a non-positive real part, are related to the short-time
behaviour of $\bar{\cal J}_n(\mu;t)$. Indeed, because of the
orthogonality property of the polynomials $p_n$ with respect to
$\mu$, it is easy to see that some of these poles are cancelled by
zeros of $G_n(\mu;z)$:
 \beq
   G_n(\mu;-2k) = 0, \;\; \mbox{for} \; 2k < n .
\nuq{shtime} We can use this information to obtain a precise
asymptotic of $\bar{\cal J}_n(\mu;t)$ for small times.

To the contrary, large times are governed by the singularities of
$M_n(\mu;z)$ at, or past, the divergence abscissa $\alpha_n(\mu)$.
In certain notable cases, these singularities can be mastered, and
a precise asymptotic obtained. In this section and in the next, we
develop the techniques to achieve this control. The same
techniques, in the absence of any information on the singularities
provide nonetheless estimates from above on the decay of
$\bar{J}_n(\mu;t)$.

Formally, the Mellin transform of any function $f$ can be seen as
a Fourier transform in {\em logarithmic time} $\tau = \log t$:
\beq
 {\cal M}(f;x+iy) = \int_{-\infty}^\infty
       e^{x \tau} f(e^\tau) \; e^{iy \tau} \; d \tau .
\nuq{mellog}
 We apply this equality to $f(t)=\bar{\cal
J}_n(\mu;t)$, and set
  \beq
 h_n(x,\tau):= e^{x \tau} \bar{\cal J}_n(\mu;e^\tau).
 \nuq{accan}
  Then,
 \beq
M_n(\mu;x+iy)) = {\cal F} (h_n(x,\tau)) (-y) ,
 \nuq{fur1}
 where, as usual, ${\cal F}$ indicates the Fourier transform.
So far, this is purely formal. We now make this precise.

%
 \bpr
 \label{th6.1}
 For $0 < x < \alpha_n(\mu)$,
$h_n(x,\tau)$ belongs to $L_1({\bf R},d\tau)$ and therefore
$M_n(\mu;x+iy)$ exists and belongs to $C_\infty({\bf R},y)$, the
set of continuous functions vanishing at infinity.
 \epr
 {\em Proof.} Because of Proposition \ref{lem5.1}, and of Fubini
 theorem,
 \[
      \int_{-\infty}^\infty \!\! d \tau
      |h_n(x,\tau)| =
  \int_{-\infty}^\infty \!\! d \tau
  | e^{x \tau} \int  d \mu(s) p_n(\mu;s) \mbox{sinc} (e^\tau s) | \leq
 \] \[ \leq
   \int_0^\infty \!\! dt \int d \mu(s) |p_n(\mu;s) t^{x-1} \mbox{sinc} (st) | =
 I_n (\mu;x) < \infty.
 \]
%

 \ble
 \label{lem6.2}
For $x < \alpha_n (\mu)$, $G_n(\mu,x+iy)$ is a bounded, continuous
function of $y$.
 \ele
 {\em Proof.} Clearly, $|G_n(\mu,x+iy)| \leq G_n(\mu;x) < \infty$,
 which also allows to apply the
dominated convergence theorem. \qed

Now, we present sufficient conditions for $M_n(\mu;x+iy))$ to
belong to a integrability class.
 \ble
  \label{lem6.3}
 For $0 < x < \min(\alpha_n (\mu),\frac{1}{2})$, the function
$M_n(\mu;x+iy))$ belongs to  $L_1({\bf R},dy)$;
 For $0 < x < \alpha_n(\mu)$, the function
$M_n(\mu;x+iy))$ belongs to  $L_2({\bf R},dy)$.
 \ele
 {\em Proof.}
Because of Stirling formula $\Gamma(z)= e^{-z}e^{(z-{1\over
2})\log z } \sqrt{2 \pi} \left[ 1 + O({1\over |z|})\right]$ for $z
\to \infty$, $|\arg z|< \pi$, we have that
%
$$ |\Gamma(x+iy-1)|\sim  \sqrt{2 \pi}~|y|^{-{3 \over 2}+x}e^{-{\pi
\over 2}|y|} ~~~\hbox{ for } |y|\to \infty.$$ Therefore, the
asymptotic behaviour of $H(x+iy)$, when $y \rightarrow \pm
\infty$, is
 \beq
   |H(x+iy)| \simeq  |y|^{-\frac{3}{2} + x}(1 + o(\frac{1}{|y|})) .
\nuq{asyh}
In addition, because of Lemma \ref{lem6.2},
$G_n(\mu;x+iy)$ is bounded in $y$ for fixed $x$. Because of the
previous lemma, and of Proposition \ref{th6.1}, $H(x+iy)$ is also
continuous. Then, it is integrable in $y$ when $x < \frac{1}{2}$,
and square summable for $x < 1$. \qed

We can therefore take the inverse Fourier transform of
$M_n(\mu;x+iy)$:
 \ble
 \label{lem6.4}
 For $0 < x < \min (\alpha_n (\mu),\frac{1}{2})$,
 \ele
\beq h_n(x,\tau) = \frac{1}{2 \pi}
   \int_{-\infty}^{\infty}  dy \;
  e^{-i \tau y}
  M_n(\mu;x+iy) .
\nuq{inve1}

{\em Proof.} In the $x$-range considered, the inverse transform of
$M_n(\mu;x+iy)$ exists, and belongs to $C_\infty({\bf R},\tau)$.
It therefore coincides with the continuous function $h_n(x,\tau)$.
\qed


This result allows us to obtain a (weaker) variant of Thm.
\ref{thfinal}:
 \bth
 \label{th6.2}
 For all $x$ such that $x <
\min (\alpha_n (\mu),\frac{1}{2})$, the Cesaro averages of the
F-B. function ${\cal J}_n(\mu;t)$ can be written as:
\[
\bar{\cal J}_n(\mu;t) =  t^{-x} s_n(x;t)
\]
where $s_n(x;t)$ belongs to $C_\infty({\bf R}_+,t)$.
 \eth
{\em Proof.}  Because of Lemma \ref{th6.2}, $h_n(x,\tau) \in
C_\infty({\bf R},\tau)$.
 Returning to linear time $t$, and using eq. (\ref{accan})
gives the thesis. \qed

The fact that in the above Theorem the value of $x$ is bound to be
smaller than one half, even when $\alpha_n(\mu)$ is not, is a
limitation due to the technique, rather than to the nature of the
problem, as it appears from Thm. \ref{thfinal}. First of all, the
range of the $L_1$ property of $M_n(x+iy)$ can sometimes be
extended. We shall see this in Sect. \ref{asymm}. Also, a few
other instructive ways exist to find a superior limit to the decay
exponent. In the next section, Sect. \ref{elle2}, we follow a
$L^2$ technique originally developed by Makarov, that permits to
obtain an asymptotic decay in weak form. Then, in Sect.
\ref{strong}, we employ a different technique to obtain the
asymptotic decay in strong form.



\section{$L^2$ representation of the asymptotic decay}
\label{elle2}

The asymptotic decay of $\bar{\cal J}_n(\mu;t)$ can be obtained
also in the full range $ 0 < \alpha_n (\mu)< 1 $ following the
$L^2$ technique of Makarov \cite{maka}:
 \bth
 \label{th7.1}
For all $x$ such that $x < \alpha_n(\mu)$, when $t \rightarrow
\infty$, one has \[ \bar{\cal J}_n(t) = t^{-x} s_n(x,t),
\] where $s_n(x,t)$ belongs to $L^2((0,\infty),t^{-1}dt)$.
 \eth
 {\em Proof.} We must consider the sequence of paths $\gamma_N$ defined
 as follows.
 They consist of the straight line $x+iy$, with $x <
\frac{1}{2}$, $y \in [-N,N]$, the horizontal line from $x+iN$ to
$\bar{x}+iN$, with $\min(\alpha_n (\mu),1) >  \bar{x} >
\frac{1}{2}$ and the two remaining segments needed to form a
rectangle in complex plane. Then, let us consider the function
 $m(\tau;z):= M_n(\mu;z) e^{- \tau z}$, with $\tau \in {\bf R}_+$
where the dependence on $n$ and $\mu$ is left implicit. This
function is analytic in the the strip $0 < Re(z) < \min (\alpha_n
(\mu),1)$ and therefore, its contour integral over $\gamma_N$ is
null. Moreover, it is easy to see that the integral on the
horizontal paths go to zero, as $N$ tends to infinity, thanks to
(\ref{asyh}). Taking this limit, the integral over the left
vertical line at $x$ tends to $2 \pi i$ times
 $h_n(x,\tau) e^{- \tau x}$,
 which is nothing else than $2 \pi i \bar{\cal J}_n(\mu;e^\tau)$,
independently of $x$. Therefore, also the integral on the right
vertical line, at $\bar{x}$, tends to the same limit. Expanding
the integral, this means that, pointwise in $\tau$,
  \beq
        \bar{\cal J}_n(\mu;e^\tau) = \frac{1}{2 \pi i}
   e^{- \tau \bar{x}} \lim_{N \rightarrow \infty}
          \int_{-N}^N
    d y \; e^{-i \tau y} M_n(\mu;\bar{x} + i y) .
\nuq{duno1}
 Consider now the integrals
\beq
        I_N =
          \int_{-N}^N
    d y \; e^{-i \tau y} M_n(\mu;\bar{x} + i y) .
\nuq{dunob1}
 Now, because of Lemma \ref{lem6.3}, when $\bar{x}$ belongs to
the range considered, the function $M_n(\mu;\bar{x} + i y)$
belongs to $L^2$, and therefore, the $L^2$ limit of the integral
(\ref{dunob1}) is a Fourier-Plancherel transform, call it $
q(\bar{x};\tau)$. Since there exists a subsequence $N_k$ such that
$I_N_k$ converges almost everywhere to $q(\bar{x};\tau)$, we see
immediately that
 \beq
   \bar{\cal J}_n(\mu;e^\tau) =
   e^{-\tau \bar{x}} q(\bar{x};\tau),
\nuq{duno2}
 where $q(\bar{x};\tau)$ belongs to $L^2(
(-\infty,\infty),d \tau)$. Returning now to linear time,
$t=e^\tau$, we get
 % \beq
   $    \bar{\cal J}_n(\mu;t) = t^{-\bar{x}} s_n(\bar{x};t), $
% \nuq{duno3}
 where $s_n(\bar{x};t)$ belongs to $L^2( (0,\infty),t^{-1} dt)$. \qed



\section{Strong form of Asymptotic decay}
\label{strong}

In this section, we apply a different technique to show that the
strong asymptotic form of $\bar{\cal J}_n(\mu;t)$ can be obtained
also in the full exponent range $[0,1]$. In fact, let $g(t)$ be a
Riemann integrable real function, $\bar{g}(t)$ be its Cesaro
average,
 $
\bar{g}(t) := {\cal C} (g;t)
 $
and let ${\cal M} (g,z,T) $ be the truncated Mellin transform
 \beq
  {\cal M} (g,z,T) := \int_0^T g(t) t^{z-1} dt.
 \nuq{parm1}
Define analogously ${\cal M}(\bar{g},z,T)$. When the improper Riemann
integral ${\cal M} (g,z,\infty)$ exists, it defines the Mellin
transform ${\cal M}(g,z)$. A simple calculation shows that
%
\ble
 For any bounded function $g$ and
 $z \neq 1$, $\Re (z) >0$:
 \beq
 {\cal M}(\bar{g},z,T) =  \frac{1}{z-1}
  [ T^z \bar{g}(T)  - {\cal M}(g_e,z,T) ],
  \nuq{cesf1}
  where $g_e(t) :=  (g(t) + g(-t))/2$
  \label{cesar1}
   \ele
 {\em Proof:} The equation above follows simply by integration by parts in the
definition of ${\cal M}(\bar{g},z,T)$. \qed \\
%
This lemma can be used to
obtain the asymptotic behaviour of $\bar{g}(t)$:

\ble
 Suppose that there exists $q \in {\bf R}$  such that the Mellin
transforms ${\cal M}(\bar{g},q)$ and ${\cal M}(g_e(t),q)$ exist as
convergent improper Riemann integrals. Then,
 \[
      \bar{g}(t) = o(t^{-q'})
  \]
 for all $q' < q$.
 \label{cesar2n}
   \ele
 {\em Proof:} Clearly, if the improper integrals defining ${\cal M}(\bar{g},q)$
and ${\cal M}(g_e,q)$ converge, so must $T^q \bar{g}(T)$ do for $T
\rightarrow \infty$, and therefore the lemma follows. \qed


We apply this Lemma to $g(t) = {\cal J}_n(\mu;t)$, the
non-averaged F-B. functions.

\bpr
 For $\Re(z) < \alpha_n(\mu)$,
the Mellin transform  of ${\cal J}_n(\mu;t)$ exists as a
convergent improper Riemann integral, and it can be represented as
  \beq
     M_n(\mu;z) =
     \int_0^\infty dt \; t^{z-1} {\cal J}_n(\mu;t)
     = \Gamma(z) e^{i \frac{\pi}{2} z}
 G_{n}(\mu;z) .
  \nuq{melg}
  \label{melgpr}
  \epr
%
{\em Proof.}
The following limit,
when it exists, defines the Mellin transform in Riemann
sense:
 \beq
  M_n(\mu;z)
   = \lim_{T \rightarrow \infty} \int_0^T dt \; t^{z-1}
 \int d\mu(s) e^{-it s} p_n(\mu;s).
 \nuq{melt1}
In fact, when $\Re z > 0$ integration at the leftmost limit is
controlled, since ${\cal J}_n(\mu;t)$ are limited. Moreover, the
double integral inside the limit is also an absolutely convergent
Lebesgue integral---therefore, the order of integration in eq.
(\ref{melt1}) can be inverted, and a change of variables
performed, to get
 \beq
  M_n(\mu;z)  = \lim_{T \rightarrow \infty}
   \int d\mu(s)
   p_n(\mu;s)
   \frac{1}{|s|^z} \int_0^{|s|T} u^{z-1} e^{iu} du.
  \nuq{melespo1}
We now move the limit within the integral with respect to $d\mu$,
in force of the dominated convergence theorem. Let $h_T(s)$ be the
integrand in eq. (\ref{melespo1}):
 \beq
  h_T(s) :=  p_n(\mu;s)
   \frac{1}{|s|^z} \int_0^{|s|T} u^{z-1} e^{iu} du.
  \nuq{accat1}
It is a matter of fact that there exists a summable majorant $h$
of $h_T$: this can be proven as follows.  In the first place the
integral
 $I(a) := \int_{0}^a  u^{z-1} e^{iu} du$ is a continuous function of the
upper integration limit $a$. Furthermore, as $a$ tends to
infinity, for all $z$ such that $0< \Re(z) < 1$, the integral
tends to a finite limit that is the Mellin transform of the
exponential function of imaginary argument, $M(e^{it},z)=\Gamma(z)
e^{i \pi z /2} $. Consequently, $|I(a)|$ is bounded by a constant
$C$, that depends only on $z$, and so is the companion integral in
eq. (\ref{accat1}). Summing all this together, we have
 \[
  | h_T(s)| \leq C   \frac{|p_n(\mu;s)|}{|s|^z} := h(s).
  \]
Since $\Re z < d_n(\mu)$, $h$ is summable, and we can conclude,
letting $M(e^{it},z)$ appear in eq. (\ref{melespo1}), that takes
the final form (\ref{melg}).
 \qed

Notice that replacing $e^{-its}$ by $\sinc(ts)$ in eq.
(\ref{melt1}) and following, we obtain an extension of the range
of  convergence of the integral representation, and of validity of
the formulae in Proposition \ref{lem5.2}:
 \bpr
 \label{melgprb}
 For $\Re(z) < \min\{d_n(\mu),2\}$,
the Mellin transform of $\bar{\cal J}_n(\mu;t)$ exists as a
convergent improper Riemann integral, and it can be represented as
in Proposition \ref{lem5.2}, eqs.
(\ref{fub4},\ref{melc1},\ref{melc2b2}).
  \epr
%
{\em Proof}. We need only to observe that the integral $I(a)$ in
the proof of Proposition \ref{melgpr} becomes here $I(a) :=
\int_{0}^a u^{z-1} \sinc (u) du$, whose modulus is a bounded
function of $a$ for all $0 < \Re(z) \leq 2$. \qed




\bth
 \label{thfin}
 % {\bf Theorem 6.2} \\
 Let $\alpha_n(\mu)$ be the divergence abscissa in Proposition \ref{lem5.1}.
 For all $x$ such that $x <  \alpha_n(\mu)$, when $t \rightarrow \infty$, one has
\[
\bar{\cal J}_n(\mu;t) = o (t^{-x})
\]
 \eth
{\em Proof.} It follows from the previous proposition that the
Mellin transforms of $\bar{\cal J}_n(\mu;t)$ and ${\cal
J}_n(\mu;t)$ have the same behaviour in $(0,1)$: as improper
Riemann integrals, they are both either convergent, or divergent.
Then, the theorem follows from Lemma \ref{cesar2n}. \qed

\bre {\rm
Proposition \ref{melgprb} above opens the possibility that an
extension of the range of asymptotic decay of $\bar{\cal
J}_n(\mu;t)$ might be found also when $1<d_0<2$. Indeed, the
result of Proposition \ref{melgpr} do not exclude that the Mellin
transform of ${\cal J}_n(\mu;t)$ might exist as an improper
Riemann integral also when $\Re (z) > 1$, if of course $\Re (z) <
d_0(\mu)$.
A case where the range of validity of strong decay is extended
beyond one  is worked out in Appendix.
} \ere


%

\section{Quadratic Amplitudes: Correlation Measure and Analyticity of MT}
\label{analy}

We now start the study of the quadratic  {\em amplitudes}
$A_{nm}(\mu;t)$. Part of this theory is merely a two dimensional
generalization of the previous sections, so that we shall be
concise in the exposition. From the definition (\ref{czz}), the
amplitudes can be written as:
%
\begin{equation}
 A_{nm}(\mu;t) =
 \dbleint d \mu(r) d \mu(s) p_n(\mu;r) p_m(\mu;s) \sinc (t(r-s)) .
   \label{cross}
   \end{equation}
The $\mu$ dependence of the orthogonal polynomials $p_n(\mu;s)$
and of other quantities will be left implicit at times, not to
overburden the notation. It is evident from eq. (\ref{cross}) that
$A_{nm}$ are real quantities.

We start by computing the integral representation of the Mellin
transform of the amplitudes $A_{nm}(\mu;t)$, that we have denoted
$M_{nm}(\mu;z)$.

Because of eq. (\ref{cross}), this  can  be written
 \beq
 M_{nm}(\mu;z) = \int_0^{\infty} dt  ~t^{z-1} \dbleint d \mu(r) d \mu(s)
  p_n(r) p_m(\mu;s) \sinc (t(r-s)).
 \nuq{emme1}

Before tackling $M_{nm}(\mu;z)$ directly,  we consider a companion
integral $I_{nm}(z)$ to (\ref{emme1}), as we did in Sect.
\ref{local}.  We have put, as usual, $z = x + i y$, with $x,y \in
{\bf R}$:
\[
   I_{nm}(z) := \dbleint  d \mu(r) d \mu(s) \int_0^{\infty} \! \! dt
    \;  |t^{z-1}| |p_n(r) p_m(\mu;s)|  | \sinc (t |r-s|) |
    =
    \]
 \beq = \dbleint d \mu(r) d \mu(s)
  \frac{
 |p_n(r) p_m(\mu;s)|}{|r-s|^x}
  \int_0^\infty \! \! d \xi \;  \xi^{x-1} |\sinc (\xi)|.
   \nuq{amp2}
The second equality has been obtained by a simple change of
variables. The second integral is convergent for $x \in (0,1)$.
The first defines the function
 \beq
  {\cal G}_{n,m}(\mu;x) = \dbleint d \mu(r) d \mu(s) \frac{
 |p_n(\mu;r) p_m(\mu;s)|}{|r-s|^x},
 \nuq{melc2xb2}
%
As in Sect. \ref{local}, we define
 \bde
 \label{defnm}
 The correlation dimensions of $\mu$, $d_{nm}(\mu)$, are the divergence
abscissas of the integrals ${\cal G}_{nm}(\mu;z)$, and the
asymptotic exponents $\alpha_{nm}(\mu)$ are the divergence
abscissas of the integrals $I_{nm}(\mu;z)$.
 \ede
%
As a matter of fact, they are here related in a simple way:
 \bpr
 \label{lem5.1-2}
 For all $n$, $ \alpha_{nn}(\mu) =  d_{nn}(\mu) \geq  D_2(\mu)$.
   For $n \neq m$, $d_{nm}(\mu) \geq D_2(\mu)$
   and $\alpha_{nm}(\mu) = \min \{d_{nm}(\mu),1\}$.
 \epr
  %
{\em Proof.} Part of the proof is similar to that of Prop.
\ref{lem5.1} and is therefore omitted. Write Holder's inequality
for the integral (\ref{melc2xb2}):
  \[  |{\cal G}_{n,m}(\mu;x)| \leq
  ( \int d \mu(r)
|p_n(\mu;r)|^q
  \int d \mu(s)
| p_m(\mu;s)|^q )^{1/q}
 (\dbleint \frac{d \mu(r) d \mu(s)}{|r-s|^{px}})^{1/p},
\]
with $p^{-1}+q^{-1}=1$. For any $q>1$ the simple integrals in the
above are convergent, since the measure $\mu$ possesses an
infinite sequence of orthogonal polynomials. The double integral,
on its part, is convergent for $px < D_2(\mu)$, that is, $x
<(1-1/q) D_2(\mu)$. The divergence abscissa of ${\cal
G}_{n,m}(\mu;z)$ is therefore larger than $D_2(\mu)-\veps$ for any
$\veps >0$.

Next, if $n=m$, $d_{nn}$ is the correlation dimension of the
measure $d\nu(r) = |p_n(\mu;r)| d \mu(r)$. Therefore, because of
Frostman theorem, it is smaller than, or equal to one. It then
coincides with $\alpha_{nn}$. This fact cannot be ascertained in
the case $n \neq m$. \qed

\bre {\rm Notice that in the previous proof the existence of
moments of arbitrary order is crucial to obtain the last
inequality in the thesis, even for finite $n,m$. In the proof of
the parallel Prop. \ref{lem5.1} a finite set of $n$ values did
only require the existence of a finite set of moments. The reason
behind this asymmetry is that the functions $p_n(\mu;s)
p_m(\mu;s)$ are {\em not} the orthogonal polynomials of the
correlation measure $\Omega$ of eq. (\ref{cordint}), unless, of
course, $n=m=0$. } \ere
%
 \bth
 \label{thfinal2}
 Let $\alpha_n(\mu)$ be the divergence abscissa in Def. \ref{defnm}.
 For all $x$ such that $x <  \alpha_{nm}(\mu)$, when $t \rightarrow \infty$, one has
$ A_{nm}(\mu;t) = o (t^{-x}) $
 \eth
{\em Proof.} Proceed as in the proof of Thm. \ref{thfinal}. \qed


 \bpr The integral representation (\ref{emme1})
defines an analytic function in the domain $0 < \Re (z) <
D_2(\mu)$. Moreover, in the same domain, the function $t
\rightarrow t^{z-1} A_{nm}(\mu;t)$ belongs to
$L^1([0,\infty],dt)$. Finally, the Mellin transform
$M_{nm}(\mu;z)$ can be written
  \beq
   M_{nm}(\mu;z) = G_{nm}(\mu;z)~ H(z),
 \nuq{gigi3}
 where $H(z)$ has been defined in eq. (\ref{melc1}) and where
$ G_{nm}(\mu;z)$ is defined by the integral representation
 \beq
   G_{nm}(\mu;z) := \dbleint  d\mu(r) d\mu(s) ~{p_n(\mu;r) p_m(\mu;s) \over
|r-s|^z}.
 \nuq{gigi1}
 \label{elle1a}
\epr
%
{\em Proof.} All statements are consequence of Prop.
\ref{lem5.1-2} and of Fubini theorem. The proof is similar to that
of Prop. \ref{lem5.2} and is therefore omitted. \qed

A trivial consequence  of Proposition (\ref{elle1a}) is
 \ble
 The divergence abscissa  of the Mellin transform
 $M_{00}(\mu;z)$ coincides with $D_2(\mu)$.
  \label{easy} \ele
  {\em Proof.} This follows easily from the factorization
(\ref{gigi3}), valid in the domain $ 0 < \Re e (z)< D_2(\mu)$.
Since $G_{00}(\mu;z)$ diverges at $D_2(\mu)$, and $D_2(\mu)$ is
smaller than, or equal to one, so necessarily does
$M_{00}(\mu;z)$. \qed

\section{Inversion Theorems for Quadratic Amplitudes}
\label{invqua}
%
We can now easily generalize the results of sections \ref{invmt},
and \ref{elle2} to the case of quadratic amplitudes. The analogues
of Prop. \ref{th6.1} and of Lemmas \ref{lem6.2} to \ref{lem6.4}
follow from an almost verbatim translation. We so arrive at:
%
\bth
  \label{lem2}
  For any $0<x< \min \{\frac{1}{2},\alpha_{nm}(\mu)\}$
  the crossed amplitudes can be written
  $ A_{nm}(\mu;t) =  t^{-x} s_{nm}(x;t)
$, where $s_{nm}(x;t)$ belongs to $C_\infty({\bf R}_+,t)$.
 Moreover, $t^{\bar{x}}
A_{nm}(\mu;t) \in L^2\left( {\bf R}_+, {dt \over t}\right)$ for
any $\bar{x}$, $0<\bar{x}< \alpha_{nm}(\mu)$. \eth
%
{\em Proof}: is analogous to that of Thms. \ref{th6.2},
\ref{th7.1}. \qed

Theorem \ref{lem2} is therefore straightforward generalization of
\cite{maka}, and shares the same weakness, when $\alpha_{nm}(\mu)
> 1/2$.

\section{Asymptotic Decay of Physical Amplitudes}
\label{asydpq}

When $n=m$, the quantities $A_{nm}(\mu;t)$ take on the physical
meaning of quantum amplitudes, that is, of occupation
probabilities. In this case, positivity properties apply and
permit to obtain the decay in strong form.

\begin{th}
 The Cesaro averages $A_{nn}(t)$ of the physical
probability amplitudes, have the asymptotic behaviour
  $$
   A_{nn}(t) =o\left(t^{-x} \right)~~~ t \to
 +\infty
 $$
for any $0<x<\alpha_{nn}(\mu)$, where $\alpha_{nn}(\mu)$ is the
divergence abscissa of the electrostatic energy of  $|p_n(\mu,r)|
\mu$, as defined in def. \ref{defnm}. \label{fth1}
 \end{th}
%
{\em Proof}. It is apparent from eq. (\ref{czz}) that $A_{nn}(t)$
is the Cesaro average of a {\em positive} function. Let us choose
arbitrarily a value $t=t_*$, and let $\xi = A_{nn}(t_*) > 0$. Let
us now study the behaviour of $A_{nn}(t)$ in the interval between
$t_*$ and infinity. Clearly, the fastest decay of $A_{nn}$ at zero
at zero is obtained when the integrand of the Cesaro average is
null for
$t' > t_*$. Let therefore $\tilde{A}_{nn}(t)$ be the function
defined by $\tilde{A}_{nn}(t_*) = A_{nn}(t_*)$,
 $
    \frac{d}{dt} \tilde{A}_{nn}(t) = - \frac{1}{t} \tilde{A}_{nn}(t),
 $
  for $t' > t_*$.
Obviously,
 $\tilde{A}_{nn}(t) = \frac{\xi t_*}{t}$, and
 $A_{nn}(t) \geq \tilde{A}_{nn}(t)$, for
 $t \in [t_*,\infty]$.
This fact, and Propositions \ref{lem5.1-2}, \ref{elle1a}, imply
that for $0 < x < d_{nn}(\mu)$
 \[
 \infty >   \int_0^\infty t^{x-1} A_{nn}(t) d t >
   \int_{t_*}^\infty t^{x-1} \tilde{A}_{nn}(t) d t
    = \frac{\xi t_*^x}{1-x}.
  \]
Recall now that $\xi  = A_{nn}(t_*)$: then, the thesis follows.
\qed

In the same way, positivity properties alone allow to overcome the
limitations of the technique of Sect. \ref{invqua}.
%
\ble
\label{lem3}
  Suppose that $f$ is the Cesaro average of a
 positive function $g$:
  \[
      f(t) = \frac{1}{t} \int_0^t g(s) ds, \; \; g(s) > 0.
  \]
Suppose also that $f(t) = t^{-{x}} s(t)$, where $s$ belongs to
$L^2( (0,\infty),t^{-1} dt)$, and $0<x<1$. Then,
 \[
       f(t) = t^{-x} o(t)  \;\; \mbox{as} \;\; t \rightarrow
       \infty.
 \]
\ele

{\em Proof.} We can proceed as in the proof of Thm. \ref{fth1}:
 Let us choose quite
arbitrarily a value $t=t_*$, and let $\xi = f(t_*)
> 0$, to obtain $f(t) \geq \tilde{f}(t) = \frac{\xi t_*}{t}$, $t \in
[t_*,\infty]$.  Therefore,
 \[
      \int_{t_*}^{\infty}
        s^2(t) \frac{dt}{t} \geq
       \int_{t_*}^{\infty} t^{2x-1} (\tilde{f}(t))^2 dt =
  (t_* \xi)^2 \int_{t_*}^{\infty} t^{2x-3}   dt   =
       \frac{t_*^{2x} \xi^2}{2-2x}.
\] Finally, employing the second hypothesis of the Lemma we see
that given any $M > 0$, for all $t_*$ larger than a certain $T$ we
have
 \[
  \int_{t_*}^{\infty}  s^2(t) \frac{dt}{t} \leq M ,
\] which implies that, under the same conditions, $
     \xi \leq \sqrt{M(2-2x)} t_*^{-x}
     $
Recalling now that $\xi = f(t_*)$ the thesis follows. \qed

This lemma permits us to prove in a different way Thm. \ref{fth1}
above:
%
{\em Proof.} Since physical amplitudes are obtained setting $n=m$
in eq. (\ref{cross}), the integrand is visibly a positive
function. Proposition \ref{lem2} and Lemma \ref{lem3}  then apply.
\qed
%
\bre {\rm A similar trick can be applied to the averaged F-B.
functions $\bar{\cal J}_n(\mu;t)$, relying on theorem \ref{th7.1}
and on their boundedness. Since this latter is---for this
aim---weaker than positivity, it leads to the weaker result
$\bar{\cal J}_n(\mu;t) = t^{-\frac{2}{3} x} o(t)$ as $t
\rightarrow \infty$, for any $x < \alpha_n(\mu)$. Clearly, this
estimate is of lesser significance than those obtained in Section
\ref{strong}. } \ere

\section{Asymptotic Decay of Quadratic Amplitudes}
\label{asycross}

 So far, we have been able to prove (in different ways)
the strong asymptotic behaviour of the Cesaro averages of the
physical amplitudes, that is, the diagonal case $n=m$. As to the
non-diagonal amplitudes, a weaker result has been obtained (Prop.
\ref{lem2}). A generalization of the technique of Sect.
\ref{strong} permits us to treat conveniently also the
non-diagonal entries $A_{nm}(\mu;t), n \neq m$.
%

\bpr
For
  $0< \Re(z) < \alpha_{nm}(\mu)$
  the Mellin transforms of ${\cal J}_n(\mu;t) {\cal
J}^*_m(\mu;t)$ has a convergent integral representation, and it
can be represented as
  \beq
     \int_0^\infty t^{z-1} {\cal J}_n(\mu;t) {\cal
J}^*_m(\mu;t) = \Gamma(z) e^{i \frac{\pi}{2} z}
 G_{nm}(\mu;z) .
  \nuq{melg2}
  \epr
%
\bpr
 \label{melgprb2}
 For $\Re(z) < \min\{d_{n,m}(\mu),2\}$,
the Mellin transform $M_{nm}(\mu;z)$ exists as a convergent
improper Riemann integral, and it can be represented as in
Proposition \ref{elle1a}, eqs. (\ref{melc1}),
(\ref{gigi3}),(\ref{gigi1}).
  \epr
%
{\em Proof.} It is analogous to that of Prop. \ref{melgpr}. \qed


Finally, we have arrived at the:
\begin{th}
 The Cesaro averages $A_{nm}(\mu;t)$
 have the asymptotic behaviour
  $$
   A_{nm}(\mu;t) =o\left(t^{-x}\right)~~~t\to
 +\infty
 $$
for any $0<x<D_2(\mu)$, where $D_2(\mu)$ is the divergence
abscissa of the electrostatic energy of  $\mu$, as defined in def.
(\ref{def1}). \label{fth2}
 \end{th}
%
{\em Proof.} Apply lemma \ref{cesar2n} with $g(t) = {\cal
J}_n(\mu;t) {\cal J}^*_m(\mu;t)$. \qed



We have therefore completed the study of the large time
asymptotics of the Cesaro averages of F-B. functions, and of
quadratic amplitudes.


\section{Asymptotic Properties of the Mellin Transform in the Strip of
Analyticity} \label{asymm}

It now our goal to study the asymptotic behaviour of the Mellin
transform $M_n(\mu;x+iy)$ itself, for large imaginary argument in
the strip of analiticity. Indeed, since $H(z)$ does not depend on
$\mu$, and features the asymptotic behaviour in eq. (\ref{asyh}),
we are left with the problem of studying $G_n(\mu;x+iy)$, for $x <
\alpha_n(\mu)$, and large $|y|$.

We suppose in this section that the support of the measure $\mu$
is bounded, and is included in ${\bf R}_+$.

 Indeed, the function $G_n(\mu;x+iy)$ can itself   be
considered as a sort of Fourier transform, as it follows from its
definition in eq. (\ref{melc2xb}):
 \beq
 G_n(\mu;x+iy) =
 \int \; d\mu(s) \frac{p_n(\mu;s)}{  |s|^{x+iy}} =
  \int d \nu_{x}(\tau) p_n(e^\tau) e^{-i y \tau}.
 \nuq{newf2}
The  measure $\nu_{x}$ is consequently defined by Riesz
representation theorem via the equation
 \beq
   \int d \nu_{x}(\tau) f(\tau) = \int d \mu(s)  |s|^{-x} f(\log(|s|)),
 \nuq{nux}
that is required to hold for any continuous function $f$ which
vanishes at infinity.

On the support of $\nu_x$, the functions $p_n(e^\tau)$ are then
bounded by the constants $C_n = \|p_n\|_{L_\infty (d\mu)}$, and
the analysis of the previous sections is simplified by this
assumption. In particular, we have the following
 \bpr \label{propog}
 Let $\mu$ have bounded support.
 The Cesaro averages of $G_n(\mu;x+iy)$,
  have an asymptotic behaviour of the kind $o(t^{-a})$,
for any $a < \min \{d_0(\nu_x),1\}$. Here, $d_0(\nu_x)$ is the
divergence abscissa of the integral ${\cal G}_\nu(w)$,
 \beq
  {\cal G}_\nu (w)  = \int d \mu(r) |r|^{-x} | \log |r| |^{-w}.
\nuq{ldimnu}
%
 The quadratic amplitudes $|G_n(\mu;x+iy)|^2$ decay in
Cesaro average as $t^{-a}$, for any $a<D_2(\nu_x)$, where
$D_2(\nu_x)$ is the divergence abscissa of the double integral
${\cal E} (\nu_x;w)$:
 \beq
 {\cal E}(\nu_x;w) =
  \int \int d \mu(r) d \mu(s) \frac {1}
  {|r|^x |s|^x \; | \log |r| - \log |s| |^w }.
  \nuq{nlu1a}
\epr
%
{\em Proof:} Notice that, actually, $d_0(\nu_x)$ is the local
dimension of $\nu_{x}$ at $\tau=0$. In fact, this latter is
defined by the divergence abscissa of the integral
 \beq
 \int d \nu_x(\tau) \frac{1}{|\tau|^w}.
 \nuq{dl1}
 We now show that this latter integral is equal to
${\cal G}_\nu (w)$. In fact, by the Beppo Levi theorem,
 \beq
 \int d \nu_x(\tau) \frac{1}{|\tau|^w} =
 \lim_{\veps \rightarrow 0}
\int d \nu_x(\tau) \chi_\veps(\tau) \frac{1}{|\tau|^w},
 \nuq{dl2}
where $\chi_\veps(\tau)$ is a continuous function, equal to one
everywhere except in a neighborhood of zero of size $\veps$, and
null in a neighborhood of zero of size $\veps/2$. By the
representation (\ref{nux}), the last limit is equal to
  \beq
  \lim_{\veps \rightarrow 0}
 \int d \mu(s) |s|^{-x} \frac{1}{|\log |s||^w}
\chi_\veps(\log |s|) = \int d \mu(s) |s|^{-x} \frac{1}{|\log
|s||^w}.
 \nuq{dl3}
The last equality in eq. ({\ref{dl3}) follows again via Beppo Levi
theorem. A similar argument shows that the integral $ {\cal E}
(\nu_x;w)$ is equal to the generalized electrostatic energy of the
measure $\nu_x$:
 \beq
 \int \int \frac{d \nu_x(r) d \nu_x(s)}{|r-s|^w} =
  \int \int d \mu(r) d \mu(s) \frac {1}
  {|r|^x |s|^x \; | \log |r| - \log |s| |^w }= {\cal E}(\nu_x;w).
  \nuq{nlu1ab}

Then, to conclude the proof, apply Theorem \ref{thfin}, and
Theorem \ref{fth1} (or equally well Theorem \ref{fth2}).
 \qed

We now study the generalised dimension $D_2(\nu_x)$ as a function
of $x$. A few properties of this function can be proven in full
generality:
 \bth
 \label{lem8.3}
 Let $\mu$ have bounded support.
Let $D_2(\nu_{x})$ be the electrostatic correlation dimension of
the measure $\nu_{x}$ defined via eq. (\ref{nux}), and let $0 \leq
x \leq x' < d_0(\mu)$. Then,
  \beq
    D_2(\nu_{x'}) \leq D_2(\nu_{x}).
 \nuq{d2lem1}
 Moreover,
 \beq
  \ba{lcl}
  D_2(\nu_{x}) \geq D_2(\mu) & \mbox{ for } & x \leq
  \frac{D_2(\mu)}{2}, \\
 D_2(\nu_{x}) \leq D_2(\mu) & \mbox{ for } & x \geq
  \frac{D_2(\mu)}{2}.
  \ea
  \nuq{nestima}
  \eth
%
{\em Proof.}
 The electrostatic energy ${\cal E} (\nu_x;w)$ of the
measure $\nu_{x}$ is defined by the double integral
  (\ref{nlu1a}).
The electrostatic correlation dimension is the divergence abscissa
of ${\cal E}(\nu_x;w)$, that can be computed restricting $w$ to
the real axis: $w \in {\bf R}$. Since we can assume by a proper
rescaling that  $|s|,|r| < 1$, the inequality $x  \leq x'$ implies
also $|r|^x \geq |r|^{x'}$, and similarly $|s|^x \geq |s|^{x'}$,
so that
 \beq
   {\cal E} (\nu_x;w) \leq  {\cal E} (\nu_{x'};w) .
   \nuq{nud1}
It then follows that the divergence abscissa of the second
function is smaller than that of the first: this is the inequality
 (\ref{d2lem1}).

More work is required to get the inequalities (\ref{nestima}). Let
us therefore derive a series of estimates.
 Firstly,
 \beq
    \frac{| |r| -  |s| |}{\max \{  |r|,|s| \} } \leq  | \log |r| - \log |s| |
    \leq
    \frac{||r| -  |s||}{\min \{  |r|,|s| \} } .
 \nuq{inlog1}
These inequalities enforce the following bounds:
 \beq
   {\cal E} (\nu_x;w) \leq {\cal U}(x;w) :=
    \dblint
    d \mu(r) d \mu(s) \frac { \max \{  |r|,|s| \}^w}
  {|r|^x |s|^x ||r| -  |s||^w }\;
 \nuq{inlog2}
 and
 \beq
   {\cal E} (\nu_x;w) \geq {\cal L}(x;w) :=
    \dblint
    d \mu(r) d \mu(s) \frac { \min \{  |r|,|s| \}^w}
  {|r|^x |s|^x ||r| -  |s||^w }\;
 \nuq{inlog3}
Moreover, let us compute
  \beq
     {\cal U}(x;w) - {\cal L}(x;w)  =
    \dblint
    d \mu(r) d \mu(s) \frac {  \max \{  |r|,|s| \}^w -\min \{  |r|,|s| \}^w}
  {|r|^x |s|^x ||r| -  |s||^w } .
 \nuq{inlog4}
Because of symmetry, we get
 \beq
     {\cal U}(x;w) - {\cal L}(x;w) \leq
     2 \int \int_{|s| \leq |r|} d \mu(r) d \mu(s)
 \frac { |r|^w - |s|^w}
  {|r|^x |s|^x ||r| -  |s||^w } .
 \nuq{inlog5}
It can be easily verified that, when $|s| \leq |r|$, and $0 < w
\leq 1$,  one has
 \beq
  0 \leq
  \frac { |r|^w - |s|^w}{ ||r| -  |s||^w }
   \leq 1 .
 \nuq{inlog6}
Then,
 \beq
     {\cal U}(x;w) - {\cal L}(x;w) \leq
     2 \int \int_{|s| \leq |r|} d \mu(r) d \mu(s)
    \frac {1}{|r|^x |s|^x}  \leq 2 ( \int d \mu(r) \frac {1}{|r|^x } )^2.
 \nuq{inlog7}
Since $x < d_0$, the last term at r.h.s. is finite, and so is the
difference ${\cal U}(x;w) - {\cal L}(x;w)$ for any value of $0 < w
\leq 1$. Therefore, the integrals ${\cal U}(x;w)$, ${\cal L}(x;w)$
{\em and} ${\cal E} (\nu_x;w)$ are all either convergent, or
divergent, when $0 < w \leq 1$.

Let us now work  on ${\cal U}(x;w)$. Let $M := \max \{  |r|,|s|
\}$ and $m := \min \{  |r|,|s| \}$.  Clearly,
 \beq
     {\cal U}(x;w) =
\dblint d \mu(r) d \mu(s) \frac { M^{w-x} }
  {m^x ||r| -  |s||^w } \geq
\dblint d \mu(r) d \mu(s) \frac { M^{w-2x} }
  { ||r| -  |s||^w }
   .
 \nuq{inlog71}
 If now $w < 2x$ we can bound $M^{w-2x} \geq 1$, and
 \beq
     {\cal U}(x;w)\geq
     \dblint d \mu(r) d \mu(s) \frac {1 }
  { ||r| -  |s||^w } := {\cal E} (\mu;w),
 \nuq{inlog8}
where ${\cal E} (\mu;w)$ is the electrostatic energy of the
measure $\mu$. Therefore, if $D_2(\mu)$ is less than $2x$, ${\cal
U}(x;w)$ is also divergent at $w=D_2(\mu)$, or possibly before:
the same goes for ${\cal E} (\nu_x;w)$, and proves the second
estimate in (\ref{nestima}).

Similar arguments can be used for ${\cal L}(x;w)$:
 \beq
     {\cal L}(x;w) =
\dblint d \mu(r) d \mu(s) \frac { m^{w-x} }
  {M^x ||r| -  |s||^w } \leq
\dblint d \mu(r) d \mu(s) \frac { m^{w-2x} }
  { ||r| -  |s||^w }
   .
 \nuq{inlog9}
Let us now assume that $w > 2x$, so that $m^{w-2x} \leq 1$:
 \beq
     {\cal L}(x;w)\leq
     \dblint d \mu(r) d \mu(s) \frac {1 }
  { ||r| -  |s||^w } := {\cal E}(\mu;w).
 \nuq{inlog10}
Therefore, if $D_2(\mu) < 2x$, ${\cal L}(x;w)$ is convergent for
all $w < D_2(\mu)$, and so is ${\cal E} (\nu_x;w)$. The divergence
abscissa $ D_2(\nu_{x})$ is therefore larger than $D_2(\mu)$: this
is the first estimate in (\ref{nestima}).
 \qed

A consequence of this theorem is that $ D_2(\nu_{D_2(\mu)/g2})=
D_2(\mu)$: this raises the question whether it is the case that
$D_2(\nu_{x})$ does {\em not} actually depend on $x$. If so, it
should always be equal to $D_2(\mu)$. This can be proven to be
true in a class of I.F.S. measures that we shall examine in a
successive paper.


\section{$L_1$ property of MT in the strip of analiticity} \label{ciccio}

Apart from being interesting in itself, the study of the previous
section is instrumental in establishing larger domains of validity
of the $L^1$ property of $M_n(\mu;x+iy)$ than those obtained in
Lemma \ref{lem6.3}. In fact, the content of Prop. \ref{propog} can
be used to show the following:
 \ble
 \label{lem8.2}

 For any $x < d_0(\mu)$ there exists
$\delta(x)>0$ such that the Cesaro average of $|G_n(\mu;x+iy)|$
 verifies
 \[
 \frac{1}{2t}   \int_{-t}^t dy \; |G_n(\mu;x+iy)| =
 o(t^{-\delta(x)}).
\]
The quantity $\delta(x)$ can be taken
as half of the electrostatic correlation dimension of
$D_2(\nu_x)$ of the measure $\nu_{x}$ defined in eq.
(\ref{nux}).
 \ele
 {\em Proof.} Observe that we can write
\[
    \int_{-t}^t dy \; |G_n(\mu;x+iy)|
 \leq ( \int_{-t}^t dy )^{\frac{1}{2} }
 ( \int_{-t}^t |G_n(\mu;x+iy)|^2 dy )^{\frac{1}{2} },
\]
which allows us to bound the asymptotic behaviour of the integral
of the modulus of $G$ by that of its {\em square} modulus.
Then, apply the results of Prop. \ref{propog}:
\[
\int_{-t}^t |G_n(\mu;x+iy)|^2 dy = o(t^{1-D_2(\nu_x)}) .
\]
The result of
the Lemma then follows. \qed

Our previous results on the $L^1$ property of $M_n(\mu;x+iy)$
have been based on the boundedness
of $G_n(\mu;x+iy)$. The new result permits us to conclude that:
 \bth
  \label{th8.1}
 % {\bf Theorem 8.1} \\
 Let
 $\delta(x)$ be as defined in Lemma \ref{lem8.2}.
 For all $x$ such that $x <
\min (d_0(\mu),\frac{1}{2}+\delta(x))$, when $t \rightarrow
\infty$, one has that $M_n(\mu;x+iy) \in L^1({\bf R},dy)$, and
consequently
$
\bar{\cal J}_n(\mu;t) = t^{-x} o(t).
$
 \eth

 {\em Proof.} Let us estimate the integral
  $\int_{-t}^t dy \; |H(x+iy)| |G_n(\mu;x+iy)|$.
Now, the asymptotic estimate (\ref{asyh}) must be combined with
the result of Lemma \ref{lem8.2}:
The former implies that there
exists $W>0$ so that, for $|y| > W$, $|H(x+iy)| \leq 2
|y|^{-\frac{3}{2}+x}$. Splitting then the integral above, we
obtain
\[
\int_{-t}^t dy |H(x+iy)| |G_n(\mu;x+iy)| \leq
 \mbox{const.} +
  2 \int_{t>|y|>W} dy \; |y|^{-\frac{3}{2}+x}  |G_n(\mu;x+iy)|,
\]
The last integral can be now computed by parts, obtaining
\[
 \int_{t>|y|>W} dy \; |y|^{-\frac{3}{2}+x}  |G_n(\mu;x+iy)| =
 \mbox{const.} +  o(t^{-\frac{1}{2} - \delta(x) +x}),
\]
whence the first part of the thesis follows.
The second part can be proven along the lines of Theorem
\ref{th6.2}.
\qed


Note that the range of values of $x$ is defined only in implicit
form in the above theorem. Crucial to its determination is the
correlation dimension  $D_2(\nu_{x}) = 2 \delta(x)$.

\section{Local Analysis at Arbitrary Point}
\label{loarb}

The analysis of the asymptotic behaviour of the Cesaro averages
${\cal J} _n(\mu;t)$ has brought into light the r\^ole of the
measure in the neighbourhood of the point zero. In fact, we can
place zero wherever we please, by usage of the shifted measure
  $ d \mu (\cdot - s)$, where $s$ is the size of the shift.
The orthogonal polynomials of the shifted measure are easily
obtained from those of the original measure. Indeed, their Jacobi
matrix, $J(s)$, is given by
$
              J(s) = J_{\mu} + s I,
$ where $I$ is the identity matrix, and $J_\mu$ is the Jacobi
matrix associated with the measure $\mu$. In addition, we have
that ${\cal J}_0(\mu(\cdot-s);t) = e^{-its} {\cal J}_0(\mu;t)$, a
relation that permits to obtain all the shifted zero order F-B.
functions in terms of the one with null shift. These simple
observations permits us to construct a {\em spectrum analyzer}
tunable to detect the local properties of a measure $\mu$ at all
points, much in the spirit of the wavelet analysis
\cite{wavsa,wavsa2}.

Also, an interesting relation can be drawn between the local
dimensions $d_0(\mu;r)$ and the correlation dimension $D_2(\mu)$.
In fact, let ${\cal C}(f)$ be the short-hand notation for the
function ${\cal C}(f;T)$. We have the following
 \ble
The divergence abscissa of the Mellin transform of the modulus of
the Cesaro average of a complex function $f$, call it $d_{|{\cal
C} (f)|}$, satisfies the inequality
 \beq
  d_{|{\cal C} (f)|} \geq d_{({\cal C}(|f|^2))^{1/2}}
 \nuq{leces1}
 \label{elecesn1}
 \ele
{\em Proof:} Apply Schwartz inequality  to the integral defining
the Cesaro average:
  \beq
 | \int_{-T}^T f(t) dt |^2 \leq \int_{-T}^T |f(t)|^2 dt
  \int_{-T}^T dt = 4 T^2 {\cal C}(|f|^2;T).
  \nuq{cesn2}
Hence, \beq
 |{\cal C}(f;T)| \leq {\cal C}(|f|^2;T)^\frac{1}{2},
  \nuq{cesn3}
and
 \beq
 %  | \int_0^W T^{x-1} {\cal C}(f,T) dt | \leq
  \int_0^W T^{x-1} | {\cal C}(f;T)| dt  \leq
  \int_0^W T^{x-1}  {\cal C}(|f|^2;T)^\frac{1}{2}  dt .
   \nuq{cesn4}
Then, if the rightmost term  is convergent, as $W$ tends to
infinity, so is the first. This implies the thesis. \qed

\ble Let $g$ be a positive function, bounded in $[0,1]$, and with
positive divergence abscissa of the Mellin transform, $d_g$. Then,
this latter is related to that of $|g|^2$, $d_{|g|^2}$, by the
inequality
 \beq
  d_g \geq \frac{1}{2} d_{|g|^2}
  \nuq{cesn5}
  \label{lecesn5}
\ele {\em Proof:} Apply Schwartz inequality to the integral
 $
 \int_1^T  t^{u} t^{x-1-u} g(t) dt ,
$
to get
 \beq
 |\int_1^T t^{x-1} g(t) dt|^2 \leq
 \frac{1}{2u+1}(T^{2u+1}-1)
 \int_1^T t^{2x-2u-2} |g(t)|^2 dt,
 \nuq{cesn6}
for any $u \neq - \frac{1}{2}$. This can also be read as
 \beq
 |M(g,x,1,T)|^2 \leq \frac{1}{2u+1}(T^{2u+1}-1)
 M(|g|^2,2x-2u-1,1,T),
 \nuq{cesn7}
where, as usual, $M(g,x,1,T)$ is the Mellin transform integral of
the function $g$ with upper integration limit $T$ and lower
integration limit one. We now let $T$ tend to infinity. Suppose
that $2u+1 < 0$, and put $a = -2u-1
> 0$. Then, if $|M(g,x,1,T)|$ tends to infinity, forcefully
$M(|g|^2,2x-2u-1,0,T) \geq M(|g|^2,2x-2u-1,1,T)$ tends to
infinity, too. The first condition is implied by $x>d_g$.
Therefore, this also  implies that $2x+a
> d_{|g|^2}$. Since this holds for any $a>0$ we obtain the thesis.
\qed

\bpr
 \label{promix1}
 The divergence abscissa of the Mellin transform of the
 modulus of the Cesaro average $\bar{\cal
J}_0(\mu;t):={\cal C} ({\cal J}_0(\mu);t)$ is larger than, or
equal to, one-half of the divergence abscissa of the Mellin
transform of the Cesaro average of $|{\cal J}_0(\mu;t)|^2$.
 \epr
%
{\em Proof:}   Let $g$ be the following function:
 \[
      g(t) := \left(
 {\cal C} (|{\cal J}_0(\mu;\cdot)|^2;t) \right)^{1/2}.
 \]
Lemma \ref{lecesn5} then implies that $d_g \geq \frac{1}{2}
d_{|g|^2}$, where now $|g|^2 = {\cal C} (|{\cal J}_0(\mu)|^2)$,
and therefore $d_{|g|^2} = D_2(\mu)$. In addition, using Lemma
\ref{elecesn1},
 \[
d_g = d_{({\cal C} (|{\cal J}_0(\mu)|^2) )^{1/2}}
 \leq d_{|{\cal C} ({\cal J}_0(\mu))|},
 \]
 and the thesis follows. \qed
%
\ble When $d(\mu;s) < 1$, the divergence abscissa of the Mellin
transform of the modulus of the Cesaro average of ${\cal
J}_0(\mu(\cdot-s);t)$ coincides with $d(\mu;s)$. \label{lemmodul}
\ele
%
{\em Proof:} Let for simplicity of notation $s=0$. Recall that,
when $d_0(\mu)<1$, one has $\alpha_0(\mu)=d_0(\mu)$. From the
definition of $I_0(x)$, eq. (\ref{fub1}), and letting $x <
d_0(\mu)$, we get:
\[
 I_0 (\mu;x) :=  \int d \mu(r) \int_0^\infty
    dt \; |t^{x-1} \frac{\sin tr}{tr} | \geq
 \int_0^\infty dt \; t^{x-1}  |
   \int d \mu(r)  \frac{\sin tr}{tr} | =
     \]
     \beq
      = {\cal M}
    (|\bar{J}_0(\mu);x) |)
  \geq  |
   \int_0^\infty
    dt \; t^{x-1}  \int d \mu(r) \frac{\sin tr}{tr} | = | {\cal M}
    (\bar{J}_0(\mu);x)  |
\nuq{fub21}
 The divergence abscissa of $I_0(x)$, $\alpha_0(\mu)$,
is therefore smaller than, or equal to,  the divergence abscissa
of ${\cal M} (|\bar{J}_0(\mu);x) |)$. Now we prove the reverse
inequality. Since $x < \alpha_0(\mu)$, the representation ${\cal
M} (\bar{J}_0(\mu);x) = G_0(\mu;x) H(x)$ holds. When $x$ tends to
$\alpha_0(\mu)$, $G_0(\mu;x)$ diverges, and so does $|{\cal M}
(\bar{J}_0(\mu);x)|$, and ${\cal M} (|\bar{J}_0(\mu);x) |)$ as
well. But this means that the divergence abscissa of this last
Mellin transform is smaller than, or equal to $\alpha_0(\mu)$.
\qed
 \bre {\rm
 Since the modulus of $\bar{J}_0(\mu)$ is a non--negative
 function, we can apply Thm.
 \ref{teolarge}, to assess that, when $d_0(\mu)<1$:
  \beq
   \limsup_{t \rightarrow \infty} \frac{\log
   C(|\bar{J}_0(\mu)|;t)}{\log t} = - d_0(\mu).
   \nuq{notte1}
 The result is nonetheless a bit involved, since it regards the
 Cesaro average of the modulus of a Cesaro average.
 } \ere

\bth The local dimension of $\mu$ at any point $s$, $d(\mu;s)$, is
larger than, or equal to, one-half of the electrostatic
correlation dimension of the measure $\mu$, $D_2(\mu)$:
 \beq
d(\mu;s) \geq \frac{1}{2} D_2(\mu) \;\; \forall s  \in {\bf R}.
\nuq{teomixe}
 \label{teomix}
  \eth
%
{\em Proof:} The electrostatic correlation dimension of the
measure $\mu$ is the divergence abscissa of the generalized
electrostatic energy ${\cal E}(\mu;z)$, eq. (\ref{elec}). It is
also the divergence abscissa of the Mellin transform of the Cesaro
average of $|{\cal J}_0(\mu;t)|^2$, as proven in sect. \ref{analy}
above.

The local dimensions $d(\mu;s)$ are the divergence abscissas of
the singular integrals ${\cal G}(\mu;s,z)$, eq. (\ref{melc2x}). It
follows from Lemma \ref{lemmodul} that, when $d(\mu;s) <1$ these
quantities coincide with the divergence abscissa of the Mellin
transform of the modulus of the Cesaro average of ${\cal
J}_0(\mu(\cdot-s);t) = e^{-its} {\cal J}_0(\mu;t)$.

Then, two cases must be considered. If $d(\mu;s) \geq
\frac{1}{2}$, then forcefully $d(\mu;s) \geq \frac{1}{2} D_2$,
since $D_2(\mu)$ is always smaller than, or equal to, one. On the
other hand, when $d(\mu;s) < \frac{1}{2}$ the identification of
the divergence abscissas of the spectral quantities
(\ref{elec},\ref{melc2x}) with their dynamical analogue is valid,
we can apply Proposition \ref{promix1}, and the thesis follows.
\qed

A few comments to this result are in order. First of all, the
validity for {\em all} points $s$ is to be remarked. This is to be
compared with the almost certain statement:

\bpr \label{aap}
The local dimensions of $\mu$ at $\mu$-almost all
points $s$, $d(\mu;s)$, are larger than, or equal to, the
electrostatic correlation dimension of the measure $\mu$,
$D_2(\mu)$:
 \[
d(\mu;s) \geq D_2(\mu) \;\; \mu \mbox{--a.e. } s \in {\bf R}.
\]
  \epr
%
{\em Proof}. For $\Re(z) < D_2(\mu)$ the integral ${\cal
E}(\mu;z)$ is convergent. Therefore, it is standard to show that
Fubini theorem implies that the integral ${\cal G}(\mu;s,z)$, the
potential at point $s$, is almost surely finite. This implies the
thesis. \qed

\bre {\rm Thm. \ref{teomix}, and the analysis of this paper, shed
light on the results of Strichartz and Last \cite{last}, that
assert that if a finite Borel measure on ${\bf R}$ is uniformly
$\alpha$ H\"older continuous (see \cite{last} or Sect.
\ref{secmel} for a definition) then the Cesaro average of
$|J_0(\mu;t)|^2$ (in our notation, $A_{00}(\mu;T)$), decays at
least as $T^{-\alpha}$, and conversely, if this is the case, then
$\mu$ is uniformly al least $\alpha/2$ continuous. } \ere

\bre {\rm The inequality (\ref{teomixe}), has also been proven to
hold separately for inferior and superior limit quantities within
the conventional formalism \cite{tre}. Also, the weaker result of
Prop. \ref{aap} is proven to hold. } \ere

\section{Conclusions}
\label{conc}

In this paper we have developed a variety of Mellin transform
techniques to analyze the asymptotic behaviour of the Cesaro
averages of the F-B. functions ${\cal J}_n(\mu;t)$ and of their
products. The analysis has brought to light the r\^ole of suitably
defined measure dimensions in defining the asymptotic decay, much
in line with previous results in the literature. These results are
now presented as belonging to a unified picture, that focuses on
the properties of the Mellin transform. The full potential of this
analysis is appreciated when applied to specific examples, like
the elementary ones presented in appendix to this paper, or  the
case of linear iterated functions systems, to which a companion
paper is devoted.

\section{Appendix I}
The general theory can be exemplified on a scholastic example: the
measures $d \mu(s) = s^p ds$ on $[0,1]$, with integer $p$. For
these measures, we have $d_0(\mu)=1+p$. Let us first focus on the
linear quantities studied in sections \ref{local} -- \ref{strong}.

Firstly, in the case $p=0$ one obtains the Lebesgue measure, with
$\alpha_0(\mu)=d_0(\mu)=1$. Letting  $g(t) = {\cal J}_0(\mu;t)$,
one easily computes the Fourier transform
\[
 g(t) = \frac{\sin(t)}{t} + \frac{i}{t} (\cos(t) - 1),
 \]
 the symmetric part
 \[
 g_e(t) =  \frac{\sin(t)}{t},
 \]
 and the Cesaro average
\[
 \bar{g}(t) =
 \frac {{\it Si}(t)}{t} \sim \frac{\pi}{2t}.
\]
It is apparent that $\bar{g}(t)$ behaves asymptotically as
$t^{-1}$ for large $t$. Theorem \ref{th6.2} permits to obtain {\em
at most} that the decay exponent is larger than, or equal to, one
half. Theorem \ref{th7.1}, in weak form, asserts that $ s(x,t) =
t^{x} \bar{g}(t)$  belongs to $L^2((0,\infty),t^{-1}dt)$ for all
$x$ less than one, a fact that is easily verified, and indeed, one
is the superior limit of the set of $x$ values for which the $L^2$
property is verified. Finally, the full strong asymptotic decay
can only be obtained via Theorem \ref{thfin}.

Let now $p=1$. Then, $d_0(\mu)=2$ is strictly larger than
$\alpha_0(\mu)=1$. In this case, explicit computations provide
\[
 g(t) = -{\frac {{e^{-i t}}\left (it+1\right )}{{t}^{2}
}}-\frac{1}{t^2},
 \]
 \[
 g_e(t) = \frac{1}{t^2} ( \cos (t) - 1) +  \frac{1}{t} \sin (t),
  \]
 and
\[
 \bar{g}(t) =  \frac{1}{t^2} (1- \cos (t)).
\]
The exponent of strong decay of $\bar{g}(t)$ is therefore two, and
cannot be obtained via Theorem \ref{thfin}. Yet,  as predicted by
Proposition \ref{melgprb}, for $\Re(z) < 2$, the Mellin transform
of $\bar{g}(t)$ exists as an improper Riemann integral. In
addition, observe that ${\cal M}(g_+)$ also exists as an improper
R.I. up to $\Re (z) < 2$. Then, Lemma \ref{cesar2n} asserts that
 $
      \bar{g}(t) = o(t^{-q'})
  $
 for all $q' < 2$.

Finally, let $p=2$, $d_0(\mu)=3$, $\alpha=1$. Here,
\[
 g(t) = \frac{2 \cos(t)}{t^2} + \frac{\sin(t)}{t} -
  2 \frac{\sin(t)}{t^3}
  + i [ \frac{2(1-\cos(t))}{t^3} - \frac{2 \sin(t)}{t^2} +
  \frac{\cos(t)}{t} ],
 \]
 and
\[
 \bar{g}(t) =
 \frac {\sin (t)}{t^3} - \frac{\cos(t)}{t^2}.
\]
The exponent of strong decay is still two, and it is intermediate
between $\alpha_0(\mu)$ and $d_0(\mu)$.

The case of quadratic amplitudes is simpler. Let now  $g(t) =
|{\cal J}_0(\mu;t)|^2$. Then, in the three cases considered above,
the leading behaviour of $ \bar{g}(t)$ is
 $2 \frac {{\it Si}(t)}{t}$ for $p=0$,
  $\frac{1}{3} \frac {{\it Si}(t)}{t}$ for $p=1$, and
   $\frac{2}{5} \frac {{\it Si}(t)}{t}$ for $p=2$. This is
   consistent with the fact that $D_2(\mu)$ is equal to one
in all the three cases. \\
%
\vspace{0.5cm} \\
Giorgio Mantica \\
Center for Non-linear and Complex Systems \\
Universit\'a dell'Insubria \\
Via Valleggio 11, 22100 Como Italy \\
giorgio@uninsubria.it \\
\vspace{0.5cm} \\
 Sandro Vaienti \\
 Centre de Physique Th\'eorique, Luminy,
 Marseille \\ and PHYMAT, Universit\'e de Toulon et du Var, France,
 and F\'ed\'eration de Recherche des Unit\'es de Math\'ematiques
 de Marseille \\


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