{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,2]],"date-time":"2026-01-02T07:43:39Z","timestamp":1767339819727},"reference-count":56,"publisher":"Springer Science and Business Media LLC","issue":"3-4","license":[{"start":{"date-parts":[[2011,4,2]],"date-time":"2011-04-02T00:00:00Z","timestamp":1301702400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Results. Math."],"published-print":{"date-parts":[[2011,5]]},"DOI":"10.1007\/s00025-010-0083-8","type":"journal-article","created":{"date-parts":[[2011,4,1]],"date-time":"2011-04-01T14:23:05Z","timestamp":1301667785000},"page":"359-400","source":"Crossref","is-referenced-by-count":26,"title":["The Summation Formulae of Euler\u2013Maclaurin, Abel\u2013Plana, Poisson, and their Interconnections with the Approximate Sampling Formula of Signal Analysis"],"prefix":"10.1007","volume":"59","author":[{"given":"P. L.","family":"Butzer","sequence":"first","affiliation":[]},{"given":"P. J. S. G.","family":"Ferreira","sequence":"additional","affiliation":[]},{"given":"G.","family":"Schmeisser","sequence":"additional","affiliation":[]},{"given":"R. L.","family":"Stens","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2011,4,2]]},"reference":[{"key":"83_CR1","unstructured":"Abel, N.H.: Solution de quelques probl\u00e8mes \u00e0 l\u2019aide d\u2019int\u00e9grales d\u00e9finies. In: Sylow, L., Lie, S. (eds.) \u0152uvres compl\u00e8tes d\u2019Abel, vol. I, pp. 11\u201327. Johnson, New York (Reprint of the Nouvelle \u00e9d., Christiania, 1881) (1965)"},{"key":"83_CR2","volume-title":"Mathematical Analysis","author":"T.M. Apostol","year":"1974","unstructured":"Apostol T.M.: Mathematical Analysis, 2nd edn. Addison-Wesley Publishing Company, Reading (1974)","edition":"2"},{"issue":"5","key":"83_CR3","doi-asserted-by":"publisher","first-page":"409","DOI":"10.1080\/00029890.1999.12005063","volume":"106","author":"T.M. Apostol","year":"1999","unstructured":"Apostol T.M.: An elementary view of Euler\u2019s summation formula. Am. Math. Monthly 106(5), 409\u2013418 (1999)","journal-title":"Am. Math. Monthly"},{"key":"83_CR4","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1088-7","volume-title":"Ramanujan\u2019s Notebooks Part. I","author":"B.C. Berndt","year":"1985","unstructured":"Berndt B.C.: Ramanujan\u2019s Notebooks Part. I. Springer, New York (1985)"},{"key":"83_CR5","volume-title":"Vorlesungen \u00fcber Fouriersche Integrale","author":"S. Bochner","year":"1948","unstructured":"Bochner S.: Vorlesungen \u00fcber Fouriersche Integrale. Chelsea Publishing Co., New York (1948)"},{"key":"83_CR6","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1016\/0022-247X(67)90183-7","volume":"18","author":"J.L. Brown","year":"1968","unstructured":"Brown, J.L. Jr.: On the error in reconstructing a non-bandlimited function by means of the bandpass sampling theorem. J. Math. Anal. Appl. 18, 75\u201384 [Erratum: J. Math. Anal. Appl. 21, 699 (1968)] (1967).","journal-title":"J. Math. Anal. Appl."},{"key":"83_CR7","doi-asserted-by":"publisher","unstructured":"Butzer, P.L., Ferreira, P.J.S.G., Higgins, J.R., Schmeisser, G., Stens, R.L.: The sampling theorem, Poisson\u2019s summation formula, general Parseval formula, reproducing kernel formula and the Paley\u2013Wiener theorem for bandlimited signals\u2014their interconnections. Appl. Anal. (in press). doi: 10.1080\/00036811003627567","DOI":"10.1080\/00036811003627567"},{"issue":"1\u20134","key":"83_CR8","first-page":"143","volume":"4","author":"P.L. Butzer","year":"1997","unstructured":"Butzer P.L., Gessinger A.: The approximate sampling theorem, Poisson\u2019s sum formula, a decomposition theorem for Parseval\u2019s equation and their interconnections. Ann. Numer. Math. 4(1\u20134), 143\u2013160 (1997)","journal-title":"Ann. Numer. Math."},{"key":"83_CR9","doi-asserted-by":"crossref","unstructured":"Butzer, P.L., Hauss, M.: Applications of sampling theory to combinatorial analysis, Stirling numbers, special functions and the Riemann zeta function. In: Higgins, J.R., Stens, R.L. (eds.) Sampling Theory in Fourier and Signal Analysis: Advanced Topics, vol. 2. Oxford University Press, Oxford (1999)","DOI":"10.1093\/oso\/9780198534969.003.0001"},{"issue":"2","key":"83_CR10","doi-asserted-by":"publisher","first-page":"250","DOI":"10.1016\/j.jat.2005.07.011","volume":"137","author":"P.L. Butzer","year":"2005","unstructured":"Butzer P.L., Higgins J.R., Stens R.L.: Classical and approximate sampling theorems: studies in the $${L^p({\\mathbb R})}$$ and the uniform norm. J. Approx. Theory 137(2), 250\u2013263 (2005)","journal-title":"J. Approx. Theory"},{"key":"83_CR11","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-7448-9","volume-title":"Fourier Analysis and Approximation","author":"P.L. Butzer","year":"1971","unstructured":"Butzer P.L., Nessel R.J.: Fourier Analysis and Approximation. Birkh\u00e4user Verlag, Basel, Academic Press, New York (1971)"},{"key":"83_CR12","doi-asserted-by":"crossref","unstructured":"Butzer, P.L., Ries, S., Stens, R.L.: Shannon\u2019s sampling theorem, Cauchy\u2019s integral formula, and related results. In: Butzer, P.L., Stens, R.L., Sz.-Nagy, B. (eds.) Anniversary Volume on Approximation Theory and Functional Analysis (Proc. Conf., Oberwolfach, Germany, 1983), ISNM, vol. 65, pp. 363\u2013377. Birkh\u00e4user Verlag, Basel (1984)","DOI":"10.1007\/978-3-0348-5432-0_33"},{"key":"83_CR13","unstructured":"Butzer, P.L., Schmeisser, G., Stens, R.L.: An introduction to sampling analysis. In: F. Marvasti (ed.) Nonuniform Sampling\u2014Theory and Practice, Information Technology: Transmission, Processing and Storage, pp. 17\u2013121. Kluwer Academic\/Plenum Publishers, New York, 2001"},{"issue":"1","key":"83_CR14","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1016\/S0019-9958(77)90264-9","volume":"34","author":"P.L. Butzer","year":"1977","unstructured":"Butzer P.L., Splettst\u00f6sser W.: A sampling theorem for duration-limited functions with error estimates. Information and Control 34(1), 55\u201365 (1977)","journal-title":"Information and Control"},{"key":"83_CR15","unstructured":"Butzer, P.L., Stens, R.L.: The Poisson summation formula, Whittaker\u2019s cardinal series and approximate integration. In: Ditzian, Z., Meir, A., Riemenschneider, S.D., Sharma A. (eds.) Second Edmonton Conference on Approximation Theory (Edmonton, Alta., 1982), CMS Conference Proceedings, vol. 3, pp. 19\u201336. American Mathematical Society, Providence (1983)"},{"key":"83_CR16","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1016\/0024-3795(83)90011-3","volume":"52\/53","author":"P.L. Butzer","year":"1983","unstructured":"Butzer, P.L., Stens, R.L.: The Euler\u2013MacLaurin summation formula, the sampling theorem, and approximate integration over the real axis. Linear Algebra Appl. 52\/53, 141\u2013155 (1983)","journal-title":"Linear Algebra Appl."},{"issue":"1\u20132","key":"83_CR17","first-page":"93","volume":"43","author":"B. Candelpergher","year":"1997","unstructured":"Candelpergher B., Coppo M.A., Delabaere E.: La sommation de Ramanujan (The Ramanujan summation). Enseign. Math. (2) 43(1\u20132), 93\u2013132 (1997)","journal-title":"Enseign. Math. (2)"},{"issue":"1","key":"83_CR18","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1007\/s11139-009-9166-0","volume":"21","author":"B. Candelpergher","year":"2010","unstructured":"Candelpergher B., Gadiyar H.G., Padma R.: Ramanujan summation and the exponential generating function $${\\sum^\\infty_{k=0} \\frac{z^k}{k!} \\zeta'(-k)}$$. Ramanujan J. 21(1), 99\u2013122 (2010)","journal-title":"Ramanujan J."},{"issue":"2","key":"83_CR19","doi-asserted-by":"publisher","first-page":"256","DOI":"10.1007\/BF02510213","volume":"37","author":"G. Dahlquist","year":"1997","unstructured":"Dahlquist, G.: On summation formulas due to Plana, Lindel\u00f6f and Abel, and related Gauss-Christoffel rules, I, II, III. BIT37(2), 256\u2013295 (1997), 37(4), 804\u2013832 (1997), 39(1), 51\u201378 (1999)","journal-title":"BIT"},{"key":"83_CR20","unstructured":"de La Vall\u00e9e Poussin, C.-J.: Collected Works\/\u0152uvres Scientifiques, vol. III. Edited by Butzer, P.L., Mawhin, J., Vetro, P. Acad\u00e9mie Royale de Belgique, Brussels, and Circolo Matematico di Palermo, Palermo (2004)"},{"issue":"5","key":"83_CR21","doi-asserted-by":"publisher","first-page":"324","DOI":"10.1080\/0025570X.1989.11977464","volume":"62","author":"J.P. Dowling","year":"1989","unstructured":"Dowling J.P.: The mathematics of the Casimir effect. Math. Mag. 62(5), 324\u2013331 (1989)","journal-title":"Math. Mag."},{"key":"83_CR22","unstructured":"Dym, H., McKean, H.P.: Fourier Series and Integrals. Probability and Mathematical Statistics, vol. 14. Academic Press, New York (1972)"},{"key":"83_CR23","unstructured":"Euler, L.: Methodus generalis summandi progressiones (A general method for summing series). Commentarii academiae scientiarum Petropolitanae 6, 68\u201397. Opera Omnia, ser. 1, vol. XIV, pp. 42\u201372. Presented to the St. Petersburg Academy on June 20, 1732. English translation by I. Bruce. http:\/\/www.math.dartmouth.edu\/~euler\/pages\/E025.html (1738)"},{"key":"83_CR24","unstructured":"Euler, L.: Inventio summae cuiusque seriei ex dato termino generali (Finding the sum of any series from a given general term). Commentarii academiae scientiarum Petropolitanae, 8, 9\u201322. Opera Omnia, ser. 1, vol. XIV, pp. 108\u2013123. Presented to the St. Petersburg Academy on October 13, 1735. English translation by J. Bell. http:\/\/www.math.dartmouth.edu\/~euler\/pages\/E047.html (1741)"},{"key":"83_CR25","unstructured":"Gauss, C.F.: Werke, XIII. Band, herausgegeben von der K\u00f6niglichen Gesellschaft der Wissenschaften zu G\u00f6ttingen (Collected works, vol. VIII). Teubner Verlag, Leipzig, 1900. Reprint: Georg Olms Verlag, Hildesheim, 1973"},{"issue":"4","key":"83_CR26","doi-asserted-by":"publisher","first-page":"174","DOI":"10.4171\/EM\/74","volume":"62","author":"W. Gautschi","year":"2007","unstructured":"Gautschi W.: Leonhard Eulers Umgang mit langsam konvergenten Reihen (Leonhard Euler\u2019s handling of slowly convergent series). Elem. Math. 62(4), 174\u2013183 (2007)","journal-title":"Elem. Math."},{"issue":"1","key":"83_CR27","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1137\/070702710","volume":"50","author":"W. Gautschi","year":"2008","unstructured":"Gautschi W.: Leonhard Euler: his life, the man, and his works. SIAM Rev. 50(1), 3\u201333 (2008)","journal-title":"SIAM Rev."},{"key":"83_CR28","doi-asserted-by":"crossref","unstructured":"Gautschi, W.: Alexander M. Ostrowski (1893\u20131986), His life, work, and students. In: Colbois, B., Riedtmann, C., Schroeder, V. (eds.) Schweizerische Mathematische Gesellschaft\u2013Soci\u00e9t\u00e9 Math\u00e9matique Suisse\u2013Swiss Mathematical Society 1910, pp. 257-278. EMS Publishing House, Z\u00fcrich (2010)","DOI":"10.4171\/089-1\/9"},{"key":"83_CR29","volume-title":"Divergent Series","author":"G.H. Hardy","year":"1949","unstructured":"Hardy G.H.: Divergent Series. Clarendon Press, Oxford (1949)"},{"key":"83_CR30","unstructured":"Henrici, P.: Applied and Computational Complex Analysis, vol. 1, Power Series\u2014Integration\u2014Conformal Mapping\u2014Location of Zeros. John Wiley & Sons, New York (1974)"},{"key":"83_CR31","doi-asserted-by":"crossref","unstructured":"Heuser, H.: Lehrbuch der Analysis, vol. I, 14th edn. B. G. Teubner, Stuttgart (2001)","DOI":"10.1007\/978-3-322-96827-2"},{"issue":"3","key":"83_CR32","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1007\/BF03549398","volume":"2","author":"J.R. Higgins","year":"2003","unstructured":"Higgins J.R.: Two basic formulae of Euler and their equivalence to Tschakalov\u2019s sampling theorem. Sampl. Theory Signal Image Process. 2(3), 259\u2013270 (2003)","journal-title":"Sampl. Theory Signal Image Process."},{"issue":"1","key":"83_CR33","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1007\/BF03549422","volume":"4","author":"J.R. Higgins","year":"2005","unstructured":"Higgins J.R.: Some groupings of equivalent results in analysis that include sampling principles. Sampl. Theory Signal Image Process. 4(1), 19\u201331 (2005)","journal-title":"Sampl. Theory Signal Image Process."},{"issue":"1","key":"83_CR34","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF03549505","volume":"8","author":"J.R. Higgins","year":"2009","unstructured":"Higgins J.R.: The Riemann zeta function and the sampling theorem. Sampl. Theory Signal Image Process. 8(1), 1\u201312 (2009)","journal-title":"Sampl. Theory Signal Image Process."},{"issue":"4","key":"83_CR35","first-page":"333","volume":"2","author":"J.R. Higgins","year":"2000","unstructured":"Higgins J.R., Schmeisser G., Voss J.J.: The sampling theorem and several equivalent results in analysis. J. Comput. Anal. Appl. 2(4), 333\u2013371 (2000)","journal-title":"J. Comput. Anal. Appl."},{"key":"83_CR36","unstructured":"Jacobi, C.G.J.: Gesammelte Werke, I. Band. Herausgegeben von C.W. Borchardt (Collected works, vol. I). G. Reimer, Berlin (1881)"},{"key":"83_CR37","unstructured":"Lindel\u00f6f, E.: Le Calcul des r\u00e9sidus et ses applications \u00e0 la th\u00e9orie des fonctions. Gauthier-Villars, Paris (Reprint: Chelsea Publ., New York, 1947) (1905)"},{"key":"83_CR38","unstructured":"Maclaurin, C.: A treatise of fluxions, 2 vols. T. W. and T. Ruddimans, 1742. Nabu Press, Edinburgh (Reproduction) (2010)"},{"issue":"1\u20133","key":"83_CR39","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF00328047","volume":"33","author":"S. Mills","year":"1985","unstructured":"Mills S.: The independent derivations by Euler, Leonhard and Maclaurin, Colin of the Euler\u2013Maclaurin summation formula. Arch. Hist. Exact Sci. 33(1\u20133), 1\u201313 (1985)","journal-title":"Arch. Hist. Exact Sci."},{"key":"83_CR40","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198539988.001.0001","volume-title":"The Casimir Effect and Its Applications","author":"V. Mostepanenko","year":"1997","unstructured":"Mostepanenko V., Trunov N.N.: The Casimir Effect and Its Applications. Clarendon Press, Oxford (1997)"},{"key":"83_CR41","doi-asserted-by":"crossref","unstructured":"Nikol\u2019ski\u012d, S.M.: Approximation of Functions of Several Variables and Imbedding Theorems. Die Grundlehren der Mathematischen Wissenschaften, vol. 205. Springer, New York (Translated from the Russian by John M. Danskin, Jr.) (1975)","DOI":"10.1007\/978-3-642-65711-5"},{"key":"83_CR42","unstructured":"Ostrowski, A.M.: Vorlesungen \u00fcber Differential- und Integralrechnung, vol. I\u2013III. Birkh\u00e4user Verlag, Basel, 1st edn. (1945\u20131954), 2nd edn. (1960\u20131967)"},{"key":"83_CR43","doi-asserted-by":"crossref","unstructured":"Ostrowski, A.M.: Collected Mathematical Papers, vol. 1\u20136. Birkh\u00e4user Verlag, Basel (1983\u20131985)","DOI":"10.1007\/978-3-0348-9358-9"},{"key":"83_CR44","first-page":"403","volume":"25","author":"G.A.A. Plana","year":"1820","unstructured":"Plana G.A.A.: Sur une nouvelle expression analytique des nombres Bernoulliens, propre \u00e0 exprimer en termes finis la formule g\u00e9n\u00e9rale pour la sommation des suites. Mem. Accad. Sci. Torino (1) 25, 403\u2013418 (1820)","journal-title":"Mem. Accad. Sci. Torino (1)"},{"key":"83_CR45","unstructured":"Prellberg, T.: The mathematics of the Casimir effect, 96 p. Annual Lectures, School of Mathematical Sciences, Queen Mary, Univ. of London, London (2007)"},{"key":"83_CR46","doi-asserted-by":"crossref","unstructured":"Rahman, Q.I., Schmeisser, G.: On a Gaussian quadrature formula for entire functions of exponential type. In: Collatz, L., Meinardus, G., N\u00fcrnberger, G. (eds.) Numerical Methods of Approximation Theory, vol. 8 (Proceedings of a Conference, Oberwolfach, Germany, 1986), ISNM, vol. 81, pp. 155\u2013168. Birkh\u00e4user Verlag, Basel (1987)","DOI":"10.1007\/978-3-0348-6656-9_14"},{"issue":"189","key":"83_CR47","doi-asserted-by":"publisher","first-page":"245","DOI":"10.1090\/S0025-5718-1990-0990602-8","volume":"54","author":"Q.I. Rahman","year":"1990","unstructured":"Rahman Q.I., Schmeisser G.: Quadrature formulae and functions of exponential type. Math. Comput. 54(189), 245\u2013270 (1990)","journal-title":"Math. Comput."},{"issue":"207","key":"83_CR48","doi-asserted-by":"publisher","first-page":"215","DOI":"10.1090\/S0025-5718-1994-1234427-0","volume":"63","author":"Q.I. Rahman","year":"1994","unstructured":"Rahman Q.I., Schmeisser G.: A quadrature formula for entire functions of exponential type. Math. Comput. 63(207), 215\u2013227 (1994)","journal-title":"Math. Comput."},{"key":"83_CR49","first-page":"151","volume":"28","author":"Q.I. Rahman","year":"1994","unstructured":"Rahman Q.I., Schmeisser G.: The summation formulae of Poisson, Plana, Euler\u2013Maclaurin and their relationship. J. Math. Sci., Delhi 28, 151\u2013171 (1994)","journal-title":"J. Math. Sci., Delhi"},{"key":"83_CR50","unstructured":"Saharian, A.A.: The generalized Abel\u2013Plana formula with applications to Bessel functions and Casimir effect. CERN document server, preprint hep-th\/0002239. http:\/\/cdsweb.cern.ch\/record\/428795\/files\/0002239.pdf (2000)"},{"issue":"4","key":"83_CR51","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1007\/s000130050398","volume":"73","author":"W. Schuster","year":"1999","unstructured":"Schuster W.: Ein Beweis der Funktionalgleichung der Riemannschen Zetafunktion (A proof of the functional equation of the Riemann zeta function). Arch. Math. (Basel) 73(4), 273\u2013275 (1999)","journal-title":"Arch. Math. (Basel)"},{"key":"83_CR52","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1093\/imamat\/12.1.103","volume":"12","author":"F. Stenger","year":"1973","unstructured":"Stenger F.: Integration formulae based on the trapezoidal formula. J. Inst. Math. Appl. 12, 103\u2013114 (1973)","journal-title":"J. Inst. Math. Appl."},{"key":"83_CR53","doi-asserted-by":"crossref","unstructured":"Terras, A.: Harmonic Analysis on Symmetric Spaces and Applications, vol. I. Springer, New York (1985)","DOI":"10.1007\/978-1-4612-5128-6"},{"key":"83_CR54","first-page":"351","volume":"10","author":"P. Weiss","year":"1963","unstructured":"Weiss P.: An estimate of the error arising from misapplication of the sampling theorem. Notices Am. Math. Soc. 10, 351 (1963)","journal-title":"Notices Am. Math. Soc."},{"key":"83_CR55","doi-asserted-by":"crossref","unstructured":"Yoshino, K.: A relation among Ramanujan\u2019s integral formula, Shannon\u2019s sampling theorem and Plana\u2019s summation formula. In: Aliyev Azero\u011flu, T., Tamrazov, P.M. (eds.) Complex Analysis and Potential Theory (Proc. Conf. satellite to ICM 2006, Gebze, Turkey, 2006), pp. 191\u2013197. World Scientific Publishing Co., Hackensack (2007)","DOI":"10.1142\/9789812778833_0019"},{"key":"83_CR56","unstructured":"Zygmund, A.: Trigonometric Series, vols. I, II, 2nd edn. Cambridge University Press, London (reprinted with corrections and some additions) (1968)"}],"container-title":["Results in Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00025-010-0083-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00025-010-0083-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00025-010-0083-8","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00025-010-0083-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,4,5]],"date-time":"2024-04-05T17:09:53Z","timestamp":1712336993000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00025-010-0083-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,4,2]]},"references-count":56,"journal-issue":{"issue":"3-4","published-print":{"date-parts":[[2011,5]]}},"alternative-id":["83"],"URL":"https:\/\/doi.org\/10.1007\/s00025-010-0083-8","relation":{},"ISSN":["1422-6383","1420-9012"],"issn-type":[{"value":"1422-6383","type":"print"},{"value":"1420-9012","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,4,2]]}}}