Abstract
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–Lebesgue norm. An important tool is the introduction of a Mellin version of the mixed Hilbert transform and its continuity properties. Our chapter extends the analogous equivalence between the classical and the approximate sampling theorem of Fourier analysis.
Dedicated to the memory of Rowland Higgins (1935–2020) a friend, model, and mentor of us all
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
C. Bardaro, P.L. Butzer, I. Mantellini, G. Schmeisser, An Introduction to Mellin Analysis and Its Applications. Book in preparation
C. Bardaro, P.L. Butzer, R.L. Stens, G. Vinti, Approximation error of the Whittaker cardinal series in term of an averaged modulus of smoothness covering discontinous signals. J. Math. Anal. Appl. 316, 269–306 (2006)
C. Bardaro, P.L. Butzer, I. Mantellini, The exponential sampling theorem of signal analysis and the reproducing kernel formula in the Mellin transform setting. Sampl. Theory Signal Image Process. 13(1), 35–66 (2014)
C. Bardaro, P.L. Butzer, I. Mantellini, The Mellin-Parseval formula and its interconnections with the exponential sampling theorem of optical physics. Integral Transforms Spec. Funct. 27(1), 17–29 (2016)
C. Bardaro, P.L. Butzer, I. Mantellini, G. Schmeisser, On the Paley-Wiener theorem in the Mellin transform setting. J. Approx. Theory 207, 60–75 (2016)
C. Bardaro, P.L. Butzer, I. Mantellini, G. Schmeisser, Mellin analysis and its basic associated metric–applications to sampling theory. Analysis Math. 42(4), 297–321 (2016)
C. Bardaro, P.L. Butzer, I. Mantellini, G. Schmeisser, A fresh approach to the Paley-Wiener theorem for Mellin transforms and the Mellin-Hardy spaces. Math. Nachr. 290, 2759–2774 (2017)
C. Bardaro, I. Mantellini, G. Schmeisser, Exponential sampling series: convergence in Mellin-Lebesgue spaces. Results Math. 74, 119 (2019)
M. Bertero, E.R. Pike, Exponential sampling method for Laplace and other dilationally invariant transforms: I. Singular-system analysis. II. Examples in photon correlation spectroscopy and Fraunhofer diffraction. Inverse Probl. 7, 1–20, 21–41 (1991)
R.P. Boas Jr., Entire Functions (Academic, New York, 1954)
J.L. Brown Jr., On the error in reconstructing a non-bandlimited function by means of the bandpass sampling theorem. J. Math. Anal. Appl. 18, 75–84 (1967); Erratum: J. Math. Anal. Appl. 21, 699 (1968)
P.L. Butzer, S. Jansche, A direct approach to the Mellin transform. J. Fourier Anal. Appl. 3, 325–375 (1997)
P.L. Butzer, S. Jansche, The exponential sampling theorem of signal analysis. Atti Sem. Mat. Fis. Univ. Modena Suppl. 46, 99–122 (1998); Special issue dedicated to Professor Calogero Vinti
P.L. Butzer, S. Jansche, A self-contained approach to Mellin transform analysis for square integrable functions; applications. Integr. Transf. Spec. Funct. 8(3–4), 175–198 (1999)
P.L. Butzer, R.J. Nessel, Fourier Analysis and Approximation, vol. I (Academic, New York, 1971)
P.L. Butzer, J.R. Higgins, R.L. Stens, Classical and approximate sampling theorems; studies in the \(L^p(\mathbb {R} )\) and in the uniform norm. J. Approx. Theory 137, 250–263 (2005)
G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities (Cambridge University Press, Cambridge, 1934)
J.R. Higgins, Sampling Theory in Fourier and Signal Analysis: Foundations (Clarendon Press, Oxford, 1996)
M.J. Marsden, F.B. Richards, S.D. Riemenschneider, Cardinal spline interpolation operators on \(\ell ^p\) data. Indiana Univ. Math. J. 24(7), 677–689 (1975)
N. Ostrowsky, D. Sornette, P. Parker, E.R. Pike, Exponential sampling method for light scattering polydispersity analysis. Opt. Acta 28, 1059–1070 (1994)
M. Riesz, Sur les fonctions conjugées. Math. Z. 27, 218–244 (1927)
A. Zygmund, Trigonometrical Series (Dover Publs, New York, 1955)
Acknowledgements
Carlo Bardaro and Ilaria Mantellini have been partially supported by the “Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA)” of the “Istituto Nazionale di Alta Matematica (INDAM)” as well as by the project “Ricerca di Base 2019 of the University of Perugia (title: Integrazione, Approssimazione Analisi non Lineare e loro Applicazioni).”
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Bardaro, C., Butzer, P.L., Mantellini, I., Schmeisser, G., Stens, R.L. (2023). Classical and Approximate Exponential Sampling Formula: Their Interconnections in Uniform and Mellin–Lebesgue Norms. In: Casey, S.D., Dodson, M.M., Ferreira, P.J.S.G., Zayed, A. (eds) Sampling, Approximation, and Signal Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-41130-4_1
Download citation
DOI: https://doi.org/10.1007/978-3-031-41130-4_1
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-031-41129-8
Online ISBN: 978-3-031-41130-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)