On the Paley–Wiener theorem in the Mellin transform setting

Bardaro, Carlo; Butzer, Paul L.; Mantellini, Ilaria; Schmeißer, Gerhard

doi:10.1016/j.jat.2016.02.010

Cited by 30 publications

(32 citation statements)

References 17 publications

(36 reference statements)

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“…In particular, analogues of the Paley–Wiener theorem were obtained for other integral transforms. Recently, in we have proved a version for Mellin transforms, independent of the Fourier theory, by introducing the notion of a Mellin–Bernstein space, comprising all functions

f \in X_{c}^{2} : = {f : {double-struckR}^{+} \to C : f false(\cdot false) {(\cdot)}^{c - 1 / 2} \in L^{2} false(R^{+} false)}

which have an analytic extension to the Riemann surface of the (complex) logarithm and satisfy some exponential‐type condition. We gave two different approaches, one involving purely complex analysis arguments and the other one using “real” arguments based on the statement of a Mellin extension of the classical Bernstein inequality also proved in .…”

Section: Introductionmentioning

confidence: 99%

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

Bardaro

Butzer

Mantellini

et al. 2017

Mathematische Nachrichten

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Abstract. 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.

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f \in X_{c}^{2} : = {f : {double-struckR}^{+} \to C : f false(\cdot false) {(\cdot)}^{c - 1 / 2} \in L^{2} false(R^{+} false)}

Section: Introductionmentioning

confidence: 99%

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

Bardaro

Butzer

Mantellini

et al. 2017

Mathematische Nachrichten

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“…By a calculation we find that dist 1 (f, B 2 c,πT ) = π and R * πT f X ∞ c = 1/2. Hence equality occurs in (5). Using the estimates of the distance functional in Mellin-Lipschitz and Mellin-Sobolev spaces, we obtain the following results.…”

Section: Approximate Mellin Reproducing Kernel Formulamentioning

confidence: 78%

“…A Mellin version of the Paley-Wiener theorem of Fourier analysis was introduced in [5], using both complex and real approaches. Moreover, the structure of the set of Mellin band-limited functions (i.e.…”

Section: Introductionmentioning

confidence: 99%

Mellin analysis and its basic associated metric—applications to sampling theory

et al. 2016

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“…The following modified space will accomplish the desired equivalence trivially. 2 For r ∈ N 0 and α > 0, define…”

Section: Characterizations Of Distances By Function Spacesmentioning

confidence: 99%

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

Bardaro

Butzer

Mantellini

et al. 2018

Calcolo

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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.

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On the Paley–Wiener theorem in the Mellin transform setting

Cited by 30 publications

References 17 publications

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

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

Mellin analysis and its basic associated metric—applications to sampling theory

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

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