Content-Type: multipart/mixed; boundary="-------------0409301005415" This is a multi-part message in MIME format. ---------------0409301005415 Content-Type: text/plain; name="04-314.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-314.comments" 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 ---------------0409301005415 Content-Type: text/plain; name="04-314.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-314.keywords" singular measures, Fourier transform, dimensions, orthogonal polynomials, Mellin transform, asymptotic ---------------0409301005415 Content-Type: application/x-tex; name="pagen-mp1.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="pagen-mp1.tex" \documentstyle[epsfig]{article} \textwidth 125mm \textheight 185mm \parindent 8mm \frenchspacing % % personal macros % % \newcommand{\qed}{\rule{2mm}{3mm}} %\newcommand{\qed}{\qed} \newcommand{\qed}{$\Box$} \newcommand{\mez}{\mbox{$\frac{1}{2}$}} \newcommand{\sinc}{\mbox{\em sinc} } % \newcommand{\qed}{\rule{2mm}{3mm}} \newcommand{\eeq}{\end{equation}} \newcommand{\beq}{\begin{equation}} \newcommand{\nuq}[1]{\label{#1} \eeq} \newcommand{\ba}{\begin{array}} \newcommand{\ea}{\end{array}} \newcommand{\veps}{\varepsilon} \newcommand{\bx}[1]{\bibitem{#1}} \newtheorem{th}{Theorem} \newcommand{\bth}{\begin{th}} \newcommand{\eth}{\end{th}} \newtheorem{lemma}{Lemma} \newtheorem{defi}{Definition} \newtheorem{rema}{Remark} \newcommand{\ble}{\begin{lemma}} \newcommand{\ele}{\end{lemma}} \newcommand{\bde}{\begin{defi}} \newcommand{\ede}{\end{defi}} \newcommand{\bre}{\begin{rema}} \newcommand{\ere}{\end{rema}} \newcommand{\ovl}[1]{\overline{#1}} \newcommand{\und}[1]{\underline{#1}} \newcommand{\ubf}[1]{\bf {#1}} \newcommand{\hbarra}{\not h} \newcommand{\dbleint}{\int \!\!\!\!\! \int \! \!} % \newcommand{\dbleint}[1]{\int \!\!\!\!\! \int_{#1} \! \!} \newcommand{\dblint}{\int \!\!\!\!\! \int \! \!} \newcommand{\ee}{\end{equation}} \newcommand{\be}{\begin{equation}} \newtheorem{prop}{Proposition} \newcommand{\epr}{\end{prop}} \newcommand{\bpr}{\begin{prop}} % % % % end of personal macros % \def\rmd{\hbox{\rm d}} \def\rme{\hbox{\rm e}} \def\rmi{\hbox{\rm i}} % \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 $x \in {\bf R_+}$, $x 0$, 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 $00$. \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 $1 1$, 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 $0 1/2$. \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 $0 0$. 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 $0 0$, 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 $0 2x$, 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 \\ \begin{thebibliography}{99} \bibitem{achie} N.I. Akhiezer, {\em The classical moment problem and some related questions in analysys}, Hafner Pub. Co., New York, 1965. \bibitem{hunt} B. R. Hunt and V. Yu. Kaloshin, {\em How projections affect the dimension spectrum of fractal measures}, {\em Nonlinearity} {\bf 10} (1997) 1031--1046. \bibitem{yorke} T. D. Sauer and J. A. Yorke, {\em Are the dimensions of a set and its image equal under typical smooth functions?} {\em Ergod. Th. \& Dynam. Sys.} {\bf 17} (1997) 941--956. \bibitem{belli2} J. H. Zhong, J. Bellissard, and R. Mosseri, {\em Green function analysis of nergy spectra scaling properties}, {\em J. Phys.: Condens. Matter} {\bf 7} (1995) 3507--3514. \bibitem{france} G. Mantica, {\em Quantum Intermittency: Old or New Phenomenon?} {\em J. Phys. IV France} {\bf 8} (1998) 253. %\bibitem{} \bibitem{belli} Further properties of the local dimension defined in this fashion can be found in H. Schulz-Baldez and J. Bellissard, {\em Anomalous transport: a mathematical framework}, {\em Rev. Math. Phys.} {\bf 10} (1998) 1--46. \bibitem{lau1} K.S. Lau, {\em Iterated function systems with overlaps and multifractal structure}, in {\em Trends in probability and related analysis} (Taipei, 1998), 35--76, World Sci. Publishing, River Edge, NJ, 1999. \bibitem{falco} K. Falconer, {\em Fractal Geometry}, Chichester, Wiley (1990). \bibitem{last} Y. Last, {\em Quantum dynamics and decompositions of singular continuous spectra}, {\em J. Funct. Anal.} {\bf 142} (1996), 406--445. \bibitem{hof} A. Hof, {\em On scaling in relation to singular spectra}, {\em Comm. Math. Phys.} {\bf 184}, (1997) 567--577. \bibitem{tre} J.M. Barbaroux, F. Germinet, S. Tcheremchantsev, {\em Generalized fractal dimensions: equivalence and basic properties}, {\em J. Math. Pures Appl.} (9) {\bf 80} (2001) 977-1012. \bibitem{wavsa} S. Vaienti and J.M. Ghez, {\em On the wavelet analysis of multifractal sets} {\em J. Stat. Phys.} {\bf 57} (9189) 415-420. \bibitem{wavsa2} S. Vaienti, {\em A Frostman-like theorem for the wavelet transform on fractal sets}, {\em Nonlinearity} {\bf 4} (1991) 1241-1249. \bibitem{wido} H. Widom, {\em Extremal Polynomials Associated with a System of Curves in the Complex Plane}, {Adv. Math.} (1969) 127-231. \bibitem{turorl} E. Orlandini, M.C. Tesi, and G. Turchetti, {\em Meromorphic structure of the Mellin transform and short-distance behavior of correlation integrals}, {\em J. Stat. Phys.} {\bf 66} (1992) 515-533. \bibitem{ketze} R. Ketzmerick, G. Petschel, and T. Geisel, {\em Slow decay of temporal correlations in quantum systems with Cantor spectra}, {\em Phys. Rev. Lett.} {\bf 69} (1992) 695-698. \bibitem{jmbarb} J.M. Barbaroux, J.M. Combes, and R. Montcho, {\em Remarks on the relation between quantum dynamics and fractal spectra}, {\em J. Math. Anal. Appl.} {\bf 213} (1997) 698-772. \bibitem{guer1} C.A. Guerin and M. Holschneider, {\em ${\cal L}^2$-Fourier asymptotic of self-similar measures}, CPT preprint, Marseille, (1996). \bibitem{guer2} C.A. Guerin and M. Holschneider, {\em On equivalent definitions of the correlation dimension for a probability measure}, {\em J. Stat. Phys.} {\bf 86} (1997) 707-720. \bibitem{l1l2} K.S. Lau, {\em Fractal measures and mean-p variations}, {\em J. Funct. Anal.} {\bf 108} (1992) 427-457. \bibitem{l2} K.S. Lau, {\em Dimension of a family of singular Bernoulli convolutions}, {\em J. Funct. Anal.} {\bf 116} (1993) 335-358. \bibitem{lw1} K.S. Lau and J. Wang, {\em Mean quadratic variations and Fourier asymptotic of self-similar measures}, {\em Monatsh. Math.} {\bf 115} (1993) 99-132. \bibitem{str1} R. Strichartz, {\em Self-similar measures and their Fourier transforms I}, {\em Indiana U. Math. J.} {\bf 39} (1990) 797-817. \bibitem{str2} R. Strichartz, {\em Self-similar measures and their Fourier transforms II}, {\em Trans. Amer. Math. Soc.} {\bf 336} (1993) 335-361. \bibitem{str0} R. S. Strichartz, {\em Fourier asymptotics of fractal measures}, {\em J. Funct. Anal.} {\bf 89} (1990) 154-187. \bibitem{maka} K.A. Makarov, {\em Asymptotic Expansions for Fourier Transform of Singular Self-Affine Measures}, {\em J. Math. An. and App.} {\bf 186} (1994) 259-286. \bibitem{physd} G. Mantica, {\em Quantum Intermittency in Almost-Periodic Lattice Systems}, {\em Physica D} {\bf 109} (1997). 113--127. \bibitem{turca} D. Bessis, J.D. Fournier, G. Servizi, G. Turchetti, and S. Vaienti, {\em Mellin transform of correlation integrals and generalized dimension of strange sets}, {\em Phys. Rev.} {\bf A 36}, 920-928 (1987). \end{thebibliography} \end{document} ---------------0409301005415--