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Carlo Bardaro

Publications and source records attributed to Carlo Bardaro.

9 recordsLinked to original sources
arXiv2019

A residue theorem for polar analytic functions and Mellin analogues of Boas' differentiation formula and Valiron's sampling formula

In this paper, we continue the study of the polar analytic functions, a notion introduced in \cite{BBMS1} and successfully applied in Mellin analysis. Here we obtain another version of the Cauchy integral formula and a residue theorem for polar Mellin derivatives, employing the new notion of logarithmic pole. The identity theorem for polar analytic functions is also derived. As applications we obtain an analogue of Boas' differentiation formula for polar Mellin derivatives, and an extension of the classical Bernstein inequality to polar Mellin derivatives. Finally we give an analogue of the well-know Valiron sampling theorem for polar analytic functions and some its consequences.

math.CV↗
arXiv2018

Development of a complex function theory upon a new concept of polar-analytic functions; Extended version

The present article is an extended version of [6] containing new results and an updated list of references. We review the notion of polar analyticity introduced in a previous paper and succesfully applied in Mellin analysis and quadrature formulae for functions defined on the positive real axis. This appears as a simple way to describe functions which are analytic on a part of the Riemann surface of the logarithm. In this paper we launch a proposal to develop a complete complex function theory, independent of classical function theory, which is built upon the new concept of polar analyticity.

math.CV↗
arXiv2018

Quadrature formulae for the positive real axis in the setting of Mellin analysis: Sharp error estimates in terms of the Mellin distance

The general Poisson summation formula of Mellin analysis can be considered as a quadrature formula for the positive real axis with remainder. For Mellin bandlimited functions it becomes an exact quadrature formula. Our main aim is to study the speed of convergence to zero of the remainder for a function $f$ in terms of its distance from a space of Mellin bandlimited functions. The resulting estimates turn out to be of best possible order. Moreover, we characterize certain rates of convergence in terms of Mellin--Sobolev and Mellin--Hardy type spaces that contain $f$. Some numerical experiments illustrate and confirm these results.

math.NA↗
arXiv2017

A fresh approach to the Paley-Wiener theorem for Mellin transforms and the Mellin-Hardy spaces

Here we give a new approach to the Paley--Wiener theorem in a Mellin analysis setting which avoids the use of the Riemann surface of the logarithm and analytical branches and is based on new concepts of "polar-analytic function" in the Mellin setting and Mellin--Bernstein spaces. A notion of Hardy spaces in the Mellin setting is also given along with applications to exponential sampling formulas of optical physics.

math.FA↗
arXiv2016

Mellin Analysis and its distance concept. Applications to sampling theory

In this paper a notion of functional "distance" in the Mellin transform setting is introduced and a general representation formula is obtained for it. Also, a determination of the distance is given in terms of Lipschitz classes and Mellin-Sobolev spaces. Finally applications to approximate versions of certain basic relations valid for Mellin band-limited functions are studied in details.

math.FA↗
arXiv2015

On the Paley-Wiener theorem in the Mellin transform setting

In this paper we establish a version of the Paley-Wiener theorem of Fourier analysis in the frame of the Mellin transform. We provide two different proofs, one involving complex analysis arguments, namely the Riemann surface of the logarithm and Cauchy theorems, and the other one employing a Bernstein inequality here derived for Mellin derivatives.

math.CA↗
arXiv2014

The foundations of fractional Mellin transform analysis

In this article we study the basic theoretical properties of Mellin-type fractional integrals, known as generalizations of the Hadamard-type fractional integrals. We give a new approach and version, specifying their semigroup property, their domain and range. Moreover we introduce a notion of strong fractional Mellin derivatives and we study the connections with the pointwise fractional Mellin derivative, which is defined by means of Hadamard-type fractional integrals. One of the main results is a fractional version of the fundamental theorem of differential and integral calculus in the Mellin frame. In fact, in this article it will be shown that the very foundations of Mellin transform theory and the corresponding analysis are quite different to those of the Fourier transform, alone since even in the simplest non-fractional case the integral operator (i.e. the anti-differentiation operator) applied to a function f will turn out to be the $R \int_0^x f(u)du=u$ with derivative $(xd/dx)f(x)$. Thus the fundamental theorem in the Mellin sense is valid in this form, one which stands apart from the classical Newtonian integral and derivative. Among the applications two fractional order partial differential equations are studied.

math.FA↗