Carlo Bardaro

University of Perugia | UNIPG · Department of Mathematics and Computer Science

We show that the exponential sampling theorem and its approximate version for functions belonging to a Mellin inversion class are equivalent in the sense that, within the setting of Mellin analysis, each can be obtained from the other as a corollary. The approximate version is considered for both, convergence in the uniform norm and in the Mellin–L...

In this paper, we study the convergence properties of certain semi‐discrete exponential‐type sampling series in a multidimensional frame. In particular, we obtain an asymptotic formula of Voronovskaya type, which gives a precise order of approximation in the space of continuous functions, and we give some particular example illustrating the theory....

The exponential sampling formula has some limitations. By incorporating a Mellin bandlimited multiplier, we extend it to a wider class of functions with a series that converges faster. This series is a generalized exponential sampling series with some interesting properties. Moreover, under a side condition, any generalized exponential sampling ser...

In this paper we study the convergence properties of certain semi-discrete exponential-type sampling series in Mellin–Lebesgue spaces. Also we examine some examples which illustrate the theory developed.

Here we review the notion of polar analyticity introduced in a previous paper and successfully applied in Mellin analysis and for quadrature formulae over the positive real axis. This approach provides a simple way of describing functions which are analytic on a part of the Riemann surface of the logarithm. New results are also obtained.

In this paper we introduce a general class of integral operators that fix exponential functions, containing several recent modified operators of Gauss–Weierstrass, or Picard or moment type operators. Pointwise convergence theorems are studied, using a Korovkin-type theorem and a Voronovskaja-type formula is obtained.

In this paper we study boundedness properties of certain semi-discrete sampling series in Mellin–Lebesgue spaces. Also we examine some examples which illustrate the theory developed. These results pave the way to the norm-convergence of these operators

In this paper, we first recall some recent results on polar-analytic functions. Then we establish Mellin analogues of a classical interpolation of Valiron and of a derivative sampling formula. As consequences a new differentiation formula and an identity theorem in Mellin–Bernstein spaces are obtained. The main tool in the proofs is a residue theor...

This volume is dedicated to the memory of our colleague and friend Domenico Candeloro (Udine, October 18, 1951-Rome, May 3, 2019). It collects contributions from several of his students, coauthors and from other prominent mathematicians who were close to him. Many of the papers are strictly related to the topics that fascinated Mimmo mostly and on...

In this paper we give a survey about recent versions of Korovkin-type theorems for modular function spaces, a class which includes $L^p$, Orlicz, Musielak-Orlicz spaces and many others. We consider various kinds of modular convergence, using certain summability processes, like triangular matrix statistical convergence, and filter convergence (which...

In this paper we introduce the exponential sampling Durrmeyer series. We discuss pointwise and uniform convergence properties and an asymptotic formula of Voronovskaja type. Quantitative results are given, using the usual modulus of continuity for uniformly continuous functions. Some examples are also described.

We establish a general version of Cauchy’s integral formula and a residue theorem for polar-analytic functions, employing the new notion of logarithmic poles. As an application, a Boas-type differentiation formula in Mellin setting and a Bernstein-type inequality for polar Mellin derivatives are deduced.

In this paper we introduce the generalized exponential sampling series of bivariate functions and establish some pointwise and uniform convergence results, also in a quantitative form. Moreover, we study the pointwise asymptotic behaviour of the series. One of the basic tools is the Mellin--Taylor formula for bivariate functions, here introduced. A...

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...

In this paper we study norm-convergence to a function f of its generalized exponential sampling series in weighted Lebesgue spaces. Key roles are taken by a result on the norm-density of the test functions and the notion of bounded coarse variation. Some examples are described.

In this paper, we develop the concept of polar analyticity introduced in Bardaro C, et al. [A fresh approach to the Paley-Wiener theorem for Mellin transforms and the Mellin-Hardy spaces. Math Nachr. 2017;290:2759–2774] and successfully applied in Mellin analysis and in quadrature of functions defined on the positive real axis (see Bardaro C, et al...

Here 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...

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 wh...

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...

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...

There axe established some conditions for existence of solutions of a nonlinear integral equation Tf = f + g, where Τ is a convolution-type integral operator.

In this paper multivariate extension of the generalized Durrmeyer sampling type series are considered. We establish a Voronovskaja type formula and a quantitative version. Finally some particular examples are discussed.

We characterize the function space whose elements have a Mellin transform with exponential decay at infinity. This result can be seen as a generalization of the Paley–Wiener theorem for Mellin transforms. As a byproduct in a similar spirit, we also characterize spaces of functions whose distances from Mellin–Paley–Wiener spaces have a prescribed as...

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 wi...

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 wi...

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-l...

We give for generalized Durrmeyer type series and their linear combinations quantitative Voronosvskaja formulae in terms of the classical Peetre K-functional. Finally we apply the general theory to various kernels

Here we introduce a generalization of the exponential sampling series of optical physics and establish pointwise and uniform convergence theorem, also in a quantitative form. Moreover we compare the error of approximation for Mellin band-limited functions using both classical and generalized exponential sampling series.

Here we introduce a generalization of the exponential sampling series of optical physics and establish pointwise and uniform convergence theorem, also in a quantitative form. Moreover we compare the error of approximation for Mellin band-limited functions using both classical and generalized exponential sampling series.

In this paper multivariate extension of the generalized Durrmeyer sampling type series are considered. We establish a Voronovskaja type formula and a quantitative version. Finally some particular examples are discussed.

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-l...

In this short note we consider suitable linear combinations of Bochner-Riesz type multivariate sampling series, which greatly improve the order of pointwise approximation. In particular we state some asymptotic formulae of Voronovskaja type which are of interest in image reconstruction. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

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.

In this paper, we establish a Mellin version of the classical Parseval formula of Fourier analysis in the case of Mellin bandlimited functions, and its equivalence with the exponential sampling formula (ESF) of signal analysis, in which the samples are not equally spaced apart as in the classical Shannon theorem, but exponentially spaced. Two quite...

Here we state a quantitative approximation theorem by means of nets of certain modified Hadamard integrals, using iterates of moment type operators, for functions f defined over the positive real semi-axis ]0, +∞[, having Mellin derivatives. The main tool is a suitable K-functional which is compatible with the structure of the multiplicative group...

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 co...

We deal with a new type of statistical convergence for double sequences, called Ψ - A -statistical convergence, and we prove a Korovkin-type approximation theorem with respect to this type of convergence in modular spaces. Finally, we give some application to moment-type operators in Orlicz spaces.

In the present paper we introduce a new type of statistical convergence for double sequences called triangular A-statistical convergence and we show that triangular A-statistical convergence and A-statistical convergence overlap, neither contains the other. Also, we give a Korovkin-type approximation theorem using this new type of convergence. Fina...

In this article we study the basic theoretical properties of Mellin-type frac- tional 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...

The aim of the paper is to extend some results concerning univariate generalized sampling approximation to the multivariate frame. We give estimates of the approximation error of the multivariate generalized sampling series for not necessarily continuous functions in \(L^{p}(\mathbb{R}^{n})\)-norm, using the averaged modulus of smoothness of Sendov...

Here, using Mellin derivatives, a different notion of moment and a suitable modulus of continuity, we state a quantitative Voronovskaja approximation formula for a general class of Mellin convolution operators. This gives a direct approach to the study of pointwise approximation of such operators, without using the Fourier analysis and its results....

The Shannon sampling theory of signal analysis, the so-called WKSsampling theorem, which can be established by methods of Fourier analysis, plays an essential role in many elds. The aim of this paper is to study the so-called exponential sampling theorem (ESF) of optical physics and engineering in which the samples are not equally spaced apart as i...

In this paper we consider a new definition of generalized sampling type series using an approach introduced by Durrmeyer for the Bernstein polynomials. We establish an asymptotic formula for functions f with a polynomial growth and as a consequence we obtain a Voronovskaja type formula. Then we consider suitable linear combinations that provide a b...

The aim of the paper is to extend some results concerning univariate generalized sampling approximation to the multivariate frame. We give estimates of the approximation error of the multivariate generalized sampling series for not necessarily continuous functions in Lp(ℝⁿ) -norm, using the averaged modulus of smoothness of Sendov and Popov type. F...

We study pointwise approximation and asymptotic formulae for a class of Mellin-Kantorovich type integral operators, both in linear and nonlinear form.

We prove some versions of abstract Korovkin-type theorems in modular function spaces, with respect to filter convergence for linear positive operators, by considering several kinds of test functions. We give some results with respect to an axiomatic convergence, including almost convergence. An extension to non positive operators is also studied. F...

In this paper suitable linear combinations for a multivariate extension of the generalized sampling series are considered, also in the Kantorovich version. These combinations provide a better order of approximation. Finally the particular example of the Bochner-Riesz kernel is discussed.

We study the problem of approximating a real-valued function f by considering sequences of general operators of sampling type, which include both discrete and integral ones. This approach gives a unitary treatment of various kinds of classical operators, such as Urysohn integral operators, in particular convolution integrals, and generalized sampli...

We prove some versions of abstract Korovkin-type theorems in modular function spaces, with respect to filter convergence for linear positive operators, by considering several kinds of test functions. We give some results with respect to an axiomatic convergence, including almost convergence. An extension to non positive operators is also studied. F...

We give some Voronovskaja formula for linear combinations of generalized sampling operators and we furnish also a quantitative version in terms of the classical Peetre K-functional. This provides a better order of approximation in the asymptotic formula. We apply the general theory to various kernels: the Bochner-Riesz kernel, translates of B-splin...

Here we give some approximation theorems concerning pointwise convergence of non- linear m-singular integral operators of Mellin type, of the form: T [m]λ (f; x) = ∞ Kλ( z,Σ mk=1 (-1)κ-1 (m κ) f(xzκ)dz/z, x > 0; m ε N, with kernels satisfying some suitable singularity assumptions. Here Λ is a non-empty set of indices with a topology and λ0 an accum...

We study the behaviour of iterates of Mellin-Fejer type operators with respect to pointwise and uniform convergence. We introduce a new method in the construction of linear combinations of Mellin type operators using the iterated kernels. In some cases this provides a better order of approximation.

Here we give a Voronovskaja formula of high order for linear combinations of the convolution operators (G(w,r)f)(s) = integral(+infinity)(0)Sigma(r)(j=1)alpha K-j(jw)(t)f(st)dt/t, where K-w is the Mellin-Gauss-Weierstrass kernel. This kind of operator provides a better order of pointwise approximation and leads to asymptotic formulae of type lim(w...

Here we give some pointwise convergence theorems and asymptotic formulae of Voronovskaja type for a general class of Kantorovich discrete operators. Applications to the Kantorovich version of some discrete operators are given.

In this paper, we study some relations concerning the algebraic and absolute moments of the bivariate Mellin–Picard kernels, and we apply the results to obtain some asymptotic formulae for the pointwise convergence. Also, quantitative estimates are given.

Here, using Mellin derivatives and a different notion of moment, we state a Voronovskaja approximation formula for a class of Mellin–Fejer type convolution operators. This new approach gives direct and simple applications to various important specific examples.

Here we give a Voronovskaja formula of high order for linear combinations of moment type convolution operators (T(n,r)f)(s) = integral(+infinity)(0) Sigma(r)(j=1) alpha(j)M(jn)(t)f(st) dt/t, where M(n) is the moment kernel. This kind of operator provides a better order of pointwise approximation and leads to asymptotic formulae of type lim(n ->+inf...

In this paper an asymptotic formula of Voronovskaja type for a multivariate extension of the Kantorovich generalized sampling series is given. Moreover a quantitative version in terms of some moduli of smoothness is established. Finally some particular examples of kernels are discussed, as the Bochner-Riesz kernel and the multivariate splines. © 20...

In this paper we study some qualitative and quantitative versions of the Voronovskaja approximation formulae for a class of bivariate Mellin convolution operators of type (Twf)(x,y)=∫R+2Kw(tx−1,vy−1)f(t,v)dtdvtv. Moreover we apply the general theory to some particular cases leading to various asymptotic formulae and involving various differential o...

Here we give some approximation theorems concerning pointwise convergence for nets of nonlinear integral operators of the form: where the kernel (K λ)λΛ satisfies some general homogeneity assumptions. Here Λ is a nonempty set of indices provided with a topology.

Using Mellin derivatives and a different notion of moment, we study asymptotic expansions for a class of Mellin-Fejer type convolution operators. Moreover, using suitable linear combinations of these operators, we obtain high order of pointwise or uniform convergence.

Here we give a quantitative Voronovskaya formula for a class of Mellin convolution operators of type (Twf)(s) = ò0+¥ Kw(zs-1)f(z)\fracdzz.({T_w}f)(s) = {\int_0^{+\infty}} {K_w}(zs^{-1})f(z)\frac{dz}{z}. Moreover we furnish various applications to some classical operators. Mathematics Subject Classification (2010)Primary 41A35-Secondary 41A25 Key...

Here we give some quantitative versions of the Voronovskaja formula for a general class of discrete operators, not necessarily positive. Applications to various generalizations of the Szász–Mirak’jan operator and a Jackson type sampling operator are given.

There are several reasons why the classical sampling theorem is rather impractical for real life signal processing. First, the sinc-kernel is not very suitable for fast and efficient computation; it decays much too slowly. Second, in practice only a finite number N of sampled values are available, so that the representation of a signal f by the fin...

We study a class of bivariate generalized sampling operators and we give a general asymptotic formula for the pointwise convergence. Moreover we study a quantitative version.

Here we give a Voronovskaya type formula for Kantorovich gen-eralized sampling series and a corresponding quantitative version in terms of some moduli of smoothness.

We introduce a class of bivariate discrete operators, not necessarily positive, and we give a general asymptotic formula for the pointwise convergence. Applications to bivariate generalized sampling series and to some Szász-Mirak'jan type operators are given.

Here we introduce a general class of discrete operators, not necessarily positive and we give a Voronovskaya-type formula for this class. Applications to generalized sampling-type operators and to a further generalization of the classical Szász-Mirak'jan operator are given. Finally a survey on Voronovskaya's formula for classical discrete operators...

In this paper a modular version of the classical Korovkin theorem in multivariate modular function spaces is obtained and applications to some multivariate discrete and integral operators, acting in Orlicz spaces, are given.

In this paper some Voronovskaya approximation formulae for a class of Mellin convolution operators of the type $ (T_w f)(x,y) = \int_{\mathbb{R}_ + ^2 } {K_w } (tx^{ - 1} ,vy^{ - 1} )f(t,v)\frac{{dtdv}} {{tv}} $ (T_w f)(x,y) = \int_{\mathbb{R}_ + ^2 } {K_w } (tx^{ - 1} ,vy^{ - 1} )f(t,v)\frac{{dtdv}} {{tv}} are given. Moreover, various examples...

Here we give some approximation theorems concerning pointwise convergence and rate of pointwise convergence for non-convolution type linear operators of the form: with kernels satisfying some general homogeneity assumptions. Here Λ is a non-empty set of indices with a topology and λ0 an accumulation point of Λ in this topology.

InthispapermodularconvergencetheoremsinOrliczspacesformultivariateextensions of the one-dimensional moment operator are given and the order of modular convergence in modular Lipschitz classes is studied.

In this paper we give some Voronovskaya approximation formulae for a class of Mellin convolution operators of type (Tw f)(s) = ò+¥0 Kw(ts-1,f(t))\fracdtt. (T_w f)(s) = {\int^{+\infty}_{0}} K_w(ts^{-1},f(t))\frac{dt}{t}. We discuss separately the linear and nonlinear cases. Moreover we furnish various applications to some classical operators.

This paper deals with the Kantorovich version of generalized sampling series, the first one to be primarily concerned with this version. It is devoted to the study of these series on Orlicz spaces, L φ (ℝ), in the instance of irregularly spaced samples. A modular convergence theorem for functions f∈L φ (ℝ) is deduced. The convergence in L p (ℝ)-spa...

We obtain an extension of the classical Korovkin theorem in abstract modular spaces. Applications to some discrete and integral operators are discussed.

The Whittaker–Shannon–Kotel'nikov sampling theorem enables one to reconstruct signals f bandlimited to [−πW,πW] from its sampled values f(k/W), k∈Z, in terms of If f is continuous but not bandlimited, one normally considers limW→∞(SWf)(t) in the supremum-norm, together with aliasing error estimates, expressed in terms of the modulus of continuity o...

In this article we study approximation properties for the class of general integral operators of the form where G is a locally compact Hausdorff topological space, (Hw )w>0 is a net of closed subsets of G with suitable properties and, for every w>0, μ Hw is a regular measure on Hw We give pointwise, uniform and modular convergence theorems in abstr...

We get some inequalities concerning the modular distance $I^\varphi_G[Tf -f]$ for bounded functions $f:G\rightarrow \mathbb{R}.$ Here $G$ is a locally compact Hausdorff topological space provided with a regular and $\sigma$-finite measure $\mu_G,$ $I^\varphi_G$ is the modular functional generating the Orlicz spaces $L^\varphi(G)$ and $T$ is a nonli...

In this paper we study approximation properties for the class of general integral operators of the form (Twf)(s) = ∫Hw Kw(s, t, f(t))dμHw(t) s ∈ G, w > 0 where G is a locally compact Hausdorff topological space, (Hw)w>0 is a net of closed subsets of G with suitable properties and, for every w > 0, μHw is a regular measure on Hw. We give pointwise,...

The present paper is a continuation of a study on abstract sampling-type operators begun in [55]. Here, nonlinear operators are considered and applications to nonlinear sampling type operators are given.

Here a new approach to the study of approximation problems by means of a family of linear integral operators is presented. The abstract approach used, which is based mainly on the use of locally compact topological groups, allows us to give a unifying method in order to study a large class of approximation problems and, as an application to signal...

In 1903 Fredholm published his famous paper on integral equations. Since then linear integral operators have become an important tool in many areas, including the theory of Fourier series and Fourier integrals, approximation theory and summability theory, and the theory of integral and differential equations. As regards the latter, applications wer...

This paper is concerned with the variation detracting property and the rate of approximation of the Bernstein, Kantorovitch and Szász-Mirak jan operators, as well as of general singular convolution operators. These problems are studied with respect to the variation seminorm. In the instance of the Bernstein polynomials the variation detracting prop...

Here we study the approximation properties of nonlinear discrete operators of type (T w f)(s)=∑ ℤ K w (s,t n /w, f(t n /w)), where {t n } is a sequence of real numbers (nodes), and {K w } is a nonlinear kernel, satisfying new and more general singularity assumptions. In particular we obtain convergence theorems in the spaces of all uniformly contin...

Here we give an estimation of the modular convergence of translated equidis-tant Riemann sums to the integral of a function belonging to a modular space. Thus we extend some previous results by Fominykh and Kaminska-Musielak.

Here we formulate a general theory which permits us to obtain approxima-tion results for a large class of nonlinear integral operators which contain, as particular cases, the nonlinear classical integral convolution operators, the nonlinear Mellin type convolution integral operators and a nonlinear version for the "generalized" sampling type operat...