{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,27]],"date-time":"2026-01-27T21:13:12Z","timestamp":1769548392242,"version":"3.49.0"},"reference-count":33,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2012,11,5]],"date-time":"2012-11-05T00:00:00Z","timestamp":1352073600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The paper is concerned with Shannon sampling reconstruction formulae of derivatives of bandlimited signals as well as of derivatives of their Hilbert transform, and their application to Boas-type formulae for higher order derivatives. The essential aim is to extend these results to non-bandlimited signals. Basic is the fact that by these extensions aliasing error terms must now be added to the bandlimited reconstruction formulae. These errors will be estimated in terms of the distance functional just introduced by the authors for the extensions of basic relations valid for bandlimited functions to larger function spaces. This approach can be regarded as a mathematical foundation of aliasing error analysis of many applications.<\/jats:p>","DOI":"10.3390\/e14112192","type":"journal-article","created":{"date-parts":[[2012,11,5]],"date-time":"2012-11-05T11:09:37Z","timestamp":1352113777000},"page":"2192-2226","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":28,"title":["Shannon\u2019s Sampling Theorem for Bandlimited Signals and Their Hilbert Transform, Boas-Type Formulae for Higher Order Derivatives\u2014The Aliasing Error Involved by Their Extensions from Bandlimited to Non-Bandlimited Signals"],"prefix":"10.3390","volume":"14","author":[{"given":"Paul L.","family":"Butzer","sequence":"first","affiliation":[{"name":"Lehrstuhl A f\u00fcr Mathematik, RWTH Aachen University, 52056 Aachen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gerhard","family":"Schmeisser","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Erlangen-Nuernberg, 91058 Erlangen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rudolf L.","family":"Stens","sequence":"additional","affiliation":[{"name":"Lehrstuhl A f\u00fcr Mathematik, RWTH Aachen University, 52056 Aachen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2012,11,5]]},"reference":[{"key":"ref_1","unstructured":"Butzer, P.L., Schmeisser, G., and Stens, R.L. Basic relations valid for the Bernstein space \n\t\t\t\t\t\t\t                  \n\t\t\t\t\t\t\t                    \n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t                          B\n\t\t\t\t\t\t\t                          \u03c3\n\t\t\t\t\t\t\t                         p\n\t\t\t\t\t\t\t                        \n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t         and their extensions to functions from larger spaces in terms of their distances from \n\t\t\t\t\t\t\t                  \n\t\t\t\t\t\t\t                    \n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t                          B\n\t\t\t\t\t\t\t                          \u03c3\n\t\t\t\t\t\t\t                          p\n\t\t\t\t\t\t\t                        \n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t        , To be submitted for publication."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Butzer, P.L., and Nessel, R.J. (1971). Fourier Analysis and Approximation, Birkh\u00e4user.","DOI":"10.1007\/978-3-0348-7448-9"},{"key":"ref_3","unstructured":"Boas, R.P. (1954). Entire Functions, Academic Press Inc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1137\/0510016","article-title":"Cardinal-type approximations of a function and its derivatives","volume":"10","author":"Lundin","year":"1979","journal-title":"SIAM J. Math. Anal."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Butzer, P.L., and Splettst\u00f6\u00dfer, W. (1977). Approximation und Interpolation durch verallgemeinerte Abtastsummen, Westdeutscher Verlag.","DOI":"10.1007\/978-3-322-88189-2"},{"key":"ref_6","first-page":"19","article-title":"The sampling theorem and linear prediction in signal analysis","volume":"90","author":"Butzer","year":"1988","journal-title":"Jahresber. Deutsch. Math.-Verein."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Marvasti, F. (2001). Nonuniform Sampling\u2014Thery and Practice, Kluwer\/Plenum.","DOI":"10.1007\/978-1-4615-1229-5"},{"key":"ref_8","unstructured":"Sch\u00fcssler, H.W. (1983, January 12\u201316). 75 Years of Aliasing Error in the Sampling Theorem. EUSIPCO\u201383, Signal Processing: Theories and Applications, Proceedings of the 2nd European Signal Processing Conference, Erlangen, Germany."},{"key":"ref_9","first-page":"699","article-title":"On the error in reconstructing a non-bandlimited function by means of the bandpass sampling theorem","volume":"21","author":"Brown","year":"1968","journal-title":"Erratum ibid."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/S0019-9958(77)90264-9","article-title":"A sampling theorem for duration-limited functions with error estimates","volume":"34","author":"Butzer","year":"1977","journal-title":"Inf. Control"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1016\/0165-1684(83)90020-8","article-title":"A unified approach to sampling theorems for derivatives and Hilbert transforms","volume":"5","author":"Stens","year":"1983","journal-title":"Signal Process"},{"key":"ref_12","unstructured":"Bracewell, R.N. (1978). The Fourier Transform and its Applications, McGraw-Hill Book Co.. [2nd ed.]."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/PL00010750","article-title":"Charakterisierung der Systeme zur Ermittlung der Hilbert-Transformation (in German with English summary) [Characterization of systems to calculate the Hilbert-Transform]","volume":"63","author":"Boche","year":"1997","journal-title":"Forschung im Ingenieurwesen"},{"key":"ref_14","unstructured":"Hahn, S.L. (1996). Hilbert Transforms in Signal Processing, The Artech House Signal Processing Library, Artech House Inc."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Feldman, M. (2011). Hilbert Transform Applications in Mechanical Vibration, John Wiley & Sons.","DOI":"10.1002\/9781119991656"},{"key":"ref_16","first-page":"13","article-title":"Konvergenzverhalten der konjugierten Shannonschen Abtastreihe (in German with English summary) [Convergence behavior of the conjugate Shannon sampling series]","volume":"5","author":"Boche","year":"1997","journal-title":"Acta Math. Inform. Univ. Ostraviensis"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1112\/jlms\/s1-12.2.164","article-title":"The derivative of a trigonometric integral","volume":"S1\u2013S12","author":"Boas","year":"1937","journal-title":"J. Lond. Math. Soc."},{"key":"ref_18","first-page":"354","article-title":"Eine trigonometrische Interpolationsformel und einige Ungleichungen f\u00fcr Polynome","volume":"23","author":"Riesz","year":"1914","journal-title":"Jahresber. Deutsch. Math.-Verein."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1016\/j.jat.2008.04.005","article-title":"Numerical differentiation inspired by a formula of R. P. Boas","volume":"160","author":"Schmeisser","year":"2009","journal-title":"J. Approx. Theory"},{"key":"ref_20","unstructured":"Lang, S. (1968). Analysis I., Addison-Wesley Publishing Co."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Bagdasarov, S. (1998). Chebyshev Splines and Kolmogorov Inequalities, Birkh\u00e4user Verlag.","DOI":"10.1007\/978-3-0348-8808-0"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Kwong, M.K., and Zettl, A. (1992). Norm Inequalities for Derivatives and Differences, Springer-Verlag. Lecture Notes in Mathematics.","DOI":"10.1007\/BFb0090864"},{"key":"ref_23","first-page":"225","article-title":"Inequalities between norms of derivatives of arbitrary functions (Russian)","volume":"26","year":"1965","journal-title":"Acta Sci. Math. (Szeged)"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Tikhomirov, V.M. (1991). Selected Works of A. N. Kolmogorov, Volume I: Mathematics and Mechanics, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-011-3030-1"},{"key":"ref_25","unstructured":"Hardy, G.H., Littlewood, J.E., and P\u00f3lya, G. (1934). Inequalities, Cambridge Mathematical Library, Cambridge University Press."},{"key":"ref_26","unstructured":"Achieser, N.I. (1956). Theory of Approximation, Frederick Ungar Publishing Co."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Timan, A.F. (1963). Theory of Approximation of Functions of a Real Variable, A Pergamon Press Book, The Macmillan Co.","DOI":"10.1016\/B978-0-08-009929-3.50008-7"},{"key":"ref_28","unstructured":"Wagner, K.W. (1939). Operatorenrechnung und Laplacesche Transformation nebst Anwendungen in Physik und Technik, Johann Ambrosius Barth Verlag. Revised edition, 1950."},{"key":"ref_29","first-page":"117","article-title":"Karl Willi Wagner zum 65. Geburtstag","volume":"2","author":"Thoma","year":"1958","journal-title":"AE\u00dc"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1007\/BF00330405","article-title":"Heaviside\u2019s operational calculus and the attempts to rigorize it","volume":"21","author":"Luetzen","year":"1979","journal-title":"Arch. Hist. Exact Sci."},{"key":"ref_31","unstructured":"Wunsch, G. (1985). Geschichte der Systemtheorie: Dynamische Systeme und Prozesse, R. Oldenbourg Verlag."},{"key":"ref_32","unstructured":"Noll, P. Nachrichtentechnik an der TH\/TU Berlin\u2014Geschichte, Stand und Ausblick. Available online: http:\/\/www.nue.tu-berlin.de\/fileadmin\/fg97\/Ueber_uns\/Geschichte\/Dokumente\/th_nachr.pdf."},{"key":"ref_33","first-page":"32","article-title":"Ein Leben zwischen Tradition und Innovation","volume":"Volume 3","author":"Dittrich","year":"2003\/2004","journal-title":"Friedrichsdorfer Schriften"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/14\/11\/2192\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:53:18Z","timestamp":1760219598000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/14\/11\/2192"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,11,5]]},"references-count":33,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2012,11]]}},"alternative-id":["e14112192"],"URL":"https:\/\/doi.org\/10.3390\/e14112192","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,11,5]]}}}