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Applicable Analysis
An International Journal
Volume 90, 2011 - Issue 3-4: Approximation Theory and Signal Analysis, a special issue in honour of Professor Paul Leo Butzer
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Original Articles

The sampling theorem, Poisson's summation formula, general Parseval formula, reproducing kernel formula and the Paley–Wiener theorem for bandlimited signals – their interconnections

P.L. Butzer Lehrstuhl A für Mathematik, RWTH Aachen University, 52056 Aachen, Germany
,
P.J.S.G. Ferreira IEETA/DETI, Universidade de Aveiro, 3810-193 Aveiro, Portugal
,
J.R. Higgins I.H.P., 4 rue du Bary, Montclar, 11250 France
,
G. Schmeisser Department of Mathematics, University of Erlangen-Nürnberg, 91054 Erlangen, Germany
&
R.L. Stens Lehrstuhl A für Mathematik, RWTH Aachen University, 52056 Aachen, GermanyCorrespondence[email protected]
Pages 431-461 | Received 14 Nov 2009, Accepted 06 Jan 2010, Published online: 09 Oct 2010

Citations (26)

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Carlo Bardaro, Paul L. Butzer & Ilaria Mantellini. (2016) The Mellin–Parseval formula and its interconnections with the exponential sampling theorem of optical physics. Integral Transforms and Special Functions 27:1, pages 17-29.
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Articles from other publishers (25)

Marco Seracini & Gianluca Vinti. (2023) Sampling by Difference as a Method of Applying the Sampling Kantorovich Model in Digital Image Processing. Applied Sciences 13:9, pages 5594.
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Karlheinz Gröchenig. 2023. Sampling, Approximation, and Signal Analysis. Sampling, Approximation, and Signal Analysis 193 210 .
Isaac Z. Pesenson. 2023. Sampling, Approximation, and Signal Analysis. Sampling, Approximation, and Signal Analysis 69 88 .
Hans G. Feichtinger. 2023. Sampling, Approximation, and Signal Analysis. Sampling, Approximation, and Signal Analysis 211 242 .
Isaac Z. Pesenson. (2022) Sampling and interpolation for the discrete Hilbert and Kak–Hilbert transforms. Canadian Mathematical Bulletin, pages 1-16.
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Jens V. Fischer & Rudolf L. Stens. (2020) On the Reversibility of Discretization. Mathematics 8:4, pages 619.
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Hussain Al-Hammali & Adel Faridani. (2020) The zeros of a sine-type function and the peak value problem. Signal Processing 167, pages 107274.
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Holger Boche & Ullrich J. Monich. (2020) Turing Computability of Fourier Transforms of Bandlimited and Discrete Signals. IEEE Transactions on Signal Processing 68, pages 532-547.
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Carlo Bardaro, Ilaria Mantellini & Gerhard Schmeisser. (2019) Exponential Sampling Series: Convergence in Mellin–Lebesgue Spaces. Results in Mathematics 74:3.
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Ayush Bhandari & Ahmed I. Zayed. (2019) Shift-invariant and sampling spaces associated with the special affine Fourier transform. Applied and Computational Harmonic Analysis 47:1, pages 30-52.
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Dong Cheng & Kit Ian Kou. (2017) Generalized sampling expansions associated with quaternion Fourier transform. Mathematical Methods in the Applied Sciences 41:11, pages 4021-4032.
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A. Krivoshein & M. Skopina. (2016) Multivariate sampling-type approximation. Analysis and Applications 15:04, pages 521-542.
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Isaac Z. Pesenson. (2016) Shannon sampling and an inverse problem for the Schrodinger equation on combinatorial graphs. Shannon sampling and an inverse problem for the Schrodinger equation on combinatorial graphs.
Paul L. Butzer, Gerhard Schmeisser & Rudolf L. Stens. (2016) Sobolev spaces of fractional order, Lipschitz spaces, readapted modulation spaces and their interrelations; applications. Journal of Approximation Theory 212, pages 1-40.
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Yuriy Kozachenko & Andriy Olenko. (2016) Whittaker–Kotel'nikov–Shannon approximation of φ-sub-Gaussian random processes. Journal of Mathematical Analysis and Applications 443:2, pages 926-946.
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Yuriy Kozachenko & Andriy Olenko. (2016) Aliasing-Truncation Errors in Sampling Approximations of Sub-Gaussian Signals. IEEE Transactions on Information Theory 62:10, pages 5831-5838.
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Hans G. Feichtinger. 2016. New Trends in Applied Harmonic Analysis. New Trends in Applied Harmonic Analysis 301 329 .
Isaac Z. Pesenson. (2015) Sampling Formulas for Groups of Operators in Banach Spaces. Sampling Theory in Signal and Image Processing 14:1, pages 1-16.
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N. H. Bingham & Badr Missaoui. (2016) Aspects of prediction. Journal of Applied Probability 51:A, pages 189-201.
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N. H. Bingham & Badr Missaoui. (2016) Aspects of prediction. Journal of Applied Probability 51:A, pages 189-201.
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P. L. Butzer, M. M. Dodson, P. J. S. G. Ferreira, J. R. Higgins, G. Schmeisser & R. L. Stens. (2014) Seven pivotal theorems of Fourier analysis, signal analysis, numerical analysis and number theory: their interconnections. Bulletin of Mathematical Sciences 4:3, pages 481-525.
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P.L. Butzer, G. Schmeisser & R.L. Stens. (2014) The classical and approximate sampling theorems and their equivalence for entire functions of exponential type. Journal of Approximation Theory 179, pages 94-111.
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M. Skopina. (2014) Band-limited scaling and wavelet expansions. Applied and Computational Harmonic Analysis 36:1, pages 143-157.
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Paul L. Butzer, Gerhard Schmeisser & Rudolf L. Stens. (2013) Basic Relations Valid for the Bernstein Space $B^{p}_{\sigma}$ and Their Extensions to Functions from Larger Spaces with Error Estimates in Terms of Their Distances from $B^{p}_{\sigma}$. Journal of Fourier Analysis and Applications 19:2, pages 333-375.
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P. L. Butzer, P. J. S. G. Ferreira, G. Schmeisser & R. L. Stens. (2011) The Summation Formulae of Euler–Maclaurin, Abel–Plana, Poisson, and their Interconnections with the Approximate Sampling Formula of Signal Analysis. Results in Mathematics 59:3-4, pages 359-400.
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