{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,2]],"date-time":"2026-01-02T07:35:58Z","timestamp":1767339358175,"version":"3.40.4"},"reference-count":85,"publisher":"World Scientific Pub Co Pte Ltd","issue":"3","license":[{"start":{"date-parts":[[2014,10,17]],"date-time":"2014-10-17T00:00:00Z","timestamp":1413504000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Bull. Math. Sci."],"published-print":{"date-parts":[[2014,12]]},"DOI":"10.1007\/s13373-014-0057-3","type":"journal-article","created":{"date-parts":[[2014,10,16]],"date-time":"2014-10-16T03:53:29Z","timestamp":1413431609000},"page":"481-525","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Seven pivotal theorems of Fourier analysis, signal analysis, numerical analysis and number theory: their interconnections"],"prefix":"10.1142","volume":"4","author":[{"given":"P. L.","family":"Butzer","sequence":"first","affiliation":[]},{"given":"M. M.","family":"Dodson","sequence":"additional","affiliation":[]},{"given":"P. J. S. G.","family":"Ferreira","sequence":"additional","affiliation":[]},{"given":"J. R.","family":"Higgins","sequence":"additional","affiliation":[]},{"given":"G.","family":"Schmeisser","sequence":"additional","affiliation":[]},{"given":"R. L.","family":"Stens","sequence":"additional","affiliation":[]}],"member":"219","published-online":{"date-parts":[[2014,10,17]]},"reference":[{"key":"57_CR1","doi-asserted-by":"crossref","unstructured":"Aldroubi, A., Gr\u00f6chenig, K.: Nonuniform sampling and reconstruction in shift-invariant spaces. SIAM Rev. 43(4), 585\u2013620 (2001). doi: 10.1137\/S0036144501386986 (electronic)","DOI":"10.1137\/S0036144501386986"},{"key":"57_CR2","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-0999-7","volume-title":"Modular Functions and Dirichlet Series in Number Theory","author":"TM Apostol","year":"1990","unstructured":"Apostol, T.M.: Modular Functions and Dirichlet Series in Number Theory, 2nd edn. Springer, New York (1990). doi: 10.1007\/978-1-4612-0999-7","edition":"2"},{"key":"57_CR3","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1090\/S0002-9947-1950-0051437-7","volume":"68","author":"N Aronszajn","year":"1950","unstructured":"Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337\u2013404 (1950)","journal-title":"Trans. Am. Math. Soc."},{"key":"57_CR4","doi-asserted-by":"crossref","unstructured":"Beaty, M.G., Dodson, M.M.: Abstract harmonic analysis and the sampling theorem. In: Higgins, J.R., Stens, R.L. (eds.) Sampling Theory in Fourier and Signal Analysis, vol 2: Advanced Topics, chap. 10, pp. 233\u2013265. Oxford University Press, Oxford (1999)","DOI":"10.1093\/oso\/9780198534969.003.0010"},{"issue":"2","key":"57_CR5","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1007\/s00041-004-8010-6","volume":"10","author":"MG Beaty","year":"2004","unstructured":"Beaty, M.G., Dodson, M.M.: The Whittaker-Kotel\u2019nikov-Shannon theorem, spectral translates and Plancherel\u2019s formula. J. Fourier Anal. Appl. 10(2), 179\u2013199 (2004). doi: 10.1007\/s00041-004-8010-6","journal-title":"J. Fourier Anal. Appl."},{"issue":"2","key":"57_CR6","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1007\/s00041-006-6025-x","volume":"13","author":"MG Beaty","year":"2007","unstructured":"Beaty, M.G., Dodson, M.M., Eveson, S.P.: A converse to Kluv\u00e1nek\u2019s theorem. J. Fourier Anal. Appl. 13(2), 187\u2013196 (2007). doi: 10.1007\/s00041-006-6025-x","journal-title":"J. Fourier Anal. Appl."},{"issue":"1\u20132","key":"57_CR7","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1016\/j.jat.2009.02.013","volume":"160","author":"MG Beaty","year":"2009","unstructured":"Beaty, M.G., Dodson, M.M., Eveson, S.P., Higgins, J.R.: On the approximate form of Kluv\u00e1nek\u2019s theorem. J. Approx. Theory 160(1\u20132), 281\u2013303 (2009). doi: 10.1016\/j.jat.2009.02.013","journal-title":"J. Approx. Theory"},{"key":"57_CR8","volume-title":"A Brief Introduction to Theta Functions","author":"R Bellman","year":"1961","unstructured":"Bellman, R.: A Brief Introduction to Theta Functions. Holt, Rinehart and Winston, New York (1961)"},{"issue":"24","key":"57_CR9","first-page":"121","volume":"2","author":"RP Boas Jr","year":"1972","unstructured":"Boas Jr, R.P.: Summation formulas and band-limited signals. T\u00f4hoku Math. J. 2(24), 121\u2013125 (1972). doi: 10.2748\/tmj\/1178241524","journal-title":"T\u00f4hoku Math. J."},{"key":"57_CR10","volume-title":"Vorlesungen \u00fcber Fouriersche Integrale","author":"S Bochner","year":"1948","unstructured":"Bochner, S.: Vorlesungen \u00fcber Fouriersche Integrale. Chelsea Publishing Co., New York (1948)"},{"issue":"1","key":"57_CR11","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/0022-247X(67)90183-7","volume":"18","author":"JL Brown Jr","year":"1967","unstructured":"Brown Jr, J.L.: On the error in reconstructing a non-bandlimited function by means of the bandpass sampling theorem. J. Math. Anal. Appl. 18(1), 75\u201384 (1967). doi: 10.1016\/0022-247X(67)90183-7","journal-title":"J. Math. Anal. Appl."},{"key":"57_CR12","doi-asserted-by":"crossref","first-page":"699","DOI":"10.1016\/0022-247X(68)90271-0","volume":"21","author":"JL Brown Jr","year":"1968","unstructured":"Brown Jr, J.L.: Erratum to: \u201cOn the error in reconstructing a nonbandlimited function by means of the bandpass sampling theorem\u201d. J. Math. Anal. Appl. 21, 699 (1968). doi: 10.1016\/0022-247X(68)90271-0","journal-title":"J. Math. Anal. Appl."},{"key":"57_CR13","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-57823-6","volume-title":"Einf\u00fchrung in die analytische Zahlentheorie","author":"J Br\u00fcdern","year":"1995","unstructured":"Br\u00fcdern, J.: Einf\u00fchrung in die analytische Zahlentheorie. Springer, Berlin (1995)"},{"key":"57_CR14","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1016\/j.jat.2013.11.010","volume":"179","author":"PL Butzer","year":"2014","unstructured":"Butzer, P.L., Schmeisser, G., Stens, R.L.: The classical and approximate sampling theorems and their equivalence for entire functions of exponential type. J. Approx. Theory 179, 94\u2013111 (2014). doi: 10.1016\/j.jat.2013.11.010","journal-title":"J. Approx. Theory"},{"issue":"3\u20134","key":"57_CR15","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1080\/00036811003627567","volume":"90","author":"PL Butzer","year":"2011","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-Wiener theorem for bandlimited signals - their interconnections. Appl. Anal. 90(3\u20134), 431\u2013461 (2011). doi: 10.1080\/00036811003627567","journal-title":"Appl. Anal."},{"issue":"3\u20134","key":"57_CR16","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1007\/s00025-010-0083-8","volume":"59","author":"PL Butzer","year":"2011","unstructured":"Butzer, P.L., Ferreira, P.J.S.G., Schmeisser, G., Stens, R.L.: The summation formulae of Euler\u2013Maclaurin, Abel\u2013Plana, Poisson, and their interconnections with the approximate sampling formula of signal analysis. Results Math. 59(3\u20134), 359\u2013400 (2011). doi: 10.1007\/s00025-010-0083-8","journal-title":"Results Math."},{"key":"57_CR17","first-page":"100","volume-title":"SampTA\u201995","author":"PL Butzer","year":"1995","unstructured":"Butzer, P.L., Gessinger, A.: A decomposition theorem for Parseval\u2019s equation; Connections with uniform and nonuniform sampling. In: Bilinskis, I., Cain, G., Marvasti, F. (eds.) SampTA\u201995, pp. 100\u2013107. Institute of Electronics and Computer Science, Riga (1995)"},{"issue":"1\u20134","key":"57_CR18","first-page":"143","volume":"4","author":"PL 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."},{"issue":"4","key":"57_CR19","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1007\/BF02649101","volume":"3","author":"PL Butzer","year":"1997","unstructured":"Butzer, P.L., Jansche, S.: A direct approach to the Mellin transform. J. Fourier Anal. Appl. 3(4), 325\u2013376 (1997). doi: 10.1007\/BF02649101","journal-title":"J. Fourier Anal. Appl."},{"key":"57_CR20","doi-asserted-by":"crossref","unstructured":"Butzer, P.L., Nessel, R.J.: Fourier Analysis and Approximation. Academic Press, New York; Birkh\u00e4user Verlag, Basel (1971)","DOI":"10.1007\/978-3-0348-7448-9"},{"key":"57_CR21","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/978-1-4615-1229-5_2","volume-title":"Nonuniform Sampling: Theory and Practice","author":"PL Butzer","year":"2001","unstructured":"Butzer, P.L., Schmeisser, G., Stens, R.L.: An introduction to sampling analysis. In: Marvasti, F. (ed.) Nonuniform Sampling: Theory and Practice, pp. 17\u2013121. Kluwer\/Plenum, New York (2001). doi: 10.1007\/978-1-4615-1229-5"},{"key":"57_CR22","doi-asserted-by":"crossref","unstructured":"Butzer, P.L., Splettst\u00f6\u00dfer, W.: Approximation und Interpolation durch verallgemeinerte Abtastsummen. Forschungsberichte des Landes Nordrhein-Westfalen Nr. 2708. Westdeutscher, Opladen (1977)","DOI":"10.1007\/978-3-322-88189-2"},{"issue":"53","key":"57_CR23","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/0024-3795(83)90011-3","volume":"52","author":"PL 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). doi: 10.1016\/0024-3795(83)80011-1","journal-title":"Linear Algebra Appl."},{"key":"57_CR24","first-page":"19","volume-title":"Second Edmonton Conference on Approximation Theory","author":"PL Butzer","year":"1983","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, pp. 19\u201336. American Mathematical Society, Providence (1983)"},{"key":"57_CR25","volume-title":"A First Course in Harmonic Analysis","author":"A Deitmar","year":"2005","unstructured":"Deitmar, A.: A First Course in Harmonic Analysis, 2nd edn. Springer, New York (2005)","edition":"2"},{"issue":"3\u20134","key":"57_CR26","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1080\/00036811003627575","volume":"90","author":"MM Dodson","year":"2011","unstructured":"Dodson, M.M.: Approximating signals in the abstract. Appl. Anal. 90(3\u20134), 563\u2013578 (2011). doi: 10.1080\/00036811003627575","journal-title":"Appl. Anal."},{"key":"57_CR27","first-page":"85","volume":"1","author":"G Doetsch","year":"1935","unstructured":"Doetsch, G.: Summatorische Eigenschaften der Besselschen Funktionen und andere Funktionalrelationen, die mit der linearen Transformationsformel der Thetafunktion \u00e4quivalent sind. Compositio Math. 1, 85\u201397 (1935)","journal-title":"Compositio Math."},{"issue":"207","key":"57_CR28","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1090\/S0025-5718-1994-1240658-6","volume":"63","author":"A Faridani","year":"1994","unstructured":"Faridani, A.: A generalized sampling theorem for locally compact abelian groups. Math. Comp. 63(207), 307\u2013327 (1994). doi: 10.2307\/2153577","journal-title":"Math. Comp."},{"key":"57_CR29","doi-asserted-by":"crossref","first-page":"455","DOI":"10.2140\/pjm.1968.25.455","volume":"25","author":"J Feldman","year":"1968","unstructured":"Feldman, J., Greenleaf, F.P.: Existence of Borel transversals in groups. Pac. J. Math. 25, 455\u2013461 (1968)","journal-title":"Pac. J. Math."},{"key":"57_CR30","unstructured":"Ferrar, W.L.: Summation formulae and their relation to Dirichlet\u2019s series. Compositio Math. 1, 344\u2013360 (1935). http:\/\/www.numdam.org\/item?id=CM_1935__1__344_0"},{"issue":"1\u20132","key":"57_CR31","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/0304-3975(95)00002-E","volume":"144","author":"P Flajolet","year":"1995","unstructured":"Flajolet, P., Gourdon, X., Dumas, P.: Mellin transforms and asymptotics: harmonic sums. Theoret. Comput. Sci. 144(1\u20132), 3\u201358 (1995). doi: 10.1016\/0304-3975(95)00002-E","journal-title":"Theoret. Comput. Sci."},{"key":"57_CR32","doi-asserted-by":"crossref","unstructured":"Garc\u00eda, A.G., Portal, A.: Sampling in reproducing kernel Banach spaces. Mediterr. J. Math. Published online: 28 Nov 2012. doi: 10.1007\/s00009-012-0234-0","DOI":"10.1007\/s00009-012-0234-0"},{"key":"57_CR33","doi-asserted-by":"crossref","unstructured":"Hamburger, H.: \u00dcber die Riemannsche Funktionalgleichung der $$\\xi $$ \u03be -Funktion. Math. Z. 10(3\u20134), 240\u2013254 (1921). doi: 10.1007\/BF01211612","DOI":"10.1007\/BF01211612"},{"key":"57_CR34","doi-asserted-by":"crossref","unstructured":"Hamburger, H.: \u00dcber die Riemannsche Funktionalgleichung der $$\\zeta $$ \u03b6 -Funktion. Math. Z. 11(3\u20134), 224\u2013245 (1921). doi: 10.1007\/BF01203626","DOI":"10.1007\/BF01203626"},{"issue":"1","key":"57_CR35","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1007\/BF01485292","volume":"13","author":"H Hamburger","year":"1922","unstructured":"Hamburger, H.: \u00dcber die Riemannsche Funktionalgleichung der $$\\zeta $$ \u03b6 -Funktion. Math. Z. 13(1), 283\u2013311 (1922). doi: 10.1007\/BF01485292","journal-title":"Math. Z."},{"key":"57_CR36","doi-asserted-by":"crossref","unstructured":"Hamburger, H.: \u00dcber einige Beziehungen, die mit der Funktionalgleichung der Riemannschen $$\\zeta $$ \u03b6 -Funktion \u00e4quivalent sind. Math. Ann. 85(1), 129\u2013140 (1922). doi: 10.1007\/BF01449611","DOI":"10.1007\/BF01449611"},{"issue":"9\u201310","key":"57_CR37","doi-asserted-by":"crossref","first-page":"971","DOI":"10.1080\/01630560903408606","volume":"30","author":"D Han","year":"2009","unstructured":"Han, D., Nashed, M.Z., Sun, Q.: Sampling expansions in reproducing kernel Hilbert and Banach spaces. Numer. Funct. Anal. Optim. 30(9\u201310), 971\u2013987 (2009). doi: 10.1080\/01630560903408606","journal-title":"Numer. Funct. Anal. Optim."},{"key":"57_CR38","doi-asserted-by":"crossref","unstructured":"Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis, vol. I: Structure of Topological Groups. Integration Theory, Group Representations. Springer, Berlin (1963)","DOI":"10.1007\/978-3-662-00102-8_5"},{"key":"57_CR39","doi-asserted-by":"crossref","unstructured":"Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis, vol. II: Structure and Analysis for Compact Groups. Analysis on Locally Compact Abelian Groups. Springer, New York (1970)","DOI":"10.1007\/978-3-642-62008-9_3"},{"key":"57_CR40","doi-asserted-by":"crossref","unstructured":"Hewitt, E., Stromberg, K.: Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable. Springer, Berlin (1965)","DOI":"10.1007\/978-3-642-88047-6"},{"key":"57_CR41","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198596998.001.0001","volume-title":"Sampling Theory in Fourier and Signal Analysis","author":"JR Higgins","year":"1996","unstructured":"Higgins, J.R.: Sampling Theory in Fourier and Signal Analysis. Clarendon Press, Oxford (1996)"},{"key":"57_CR42","doi-asserted-by":"crossref","unstructured":"Higgins, J.R.: A sampling principle associated with Saitoh\u2019s fundamental theory of linear transformations. In: Saitoh, S., Hayashi, N., Yamamoto, M. (eds.) Analytic Extension Formulas and Their Applications (Collected papers 2nd Internat. Congress of the Internat. Society for Analysis, its Applications and Computation (ISAAC\u201999), Fukuoka, Japan, Aug. 16\u201321, 1999, and Research Meeting on Applications of Analytic Extensions, Kyoto, Japan, Jan. 11\u201313, 2000). Kluwer Academic Publishers, Dordrecht, pp. 73\u201386 (2001)","DOI":"10.1007\/978-1-4757-3298-6_6"},{"issue":"3","key":"57_CR43","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1007\/BF03549398","volume":"2","author":"JR 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."},{"key":"57_CR44","doi-asserted-by":"crossref","unstructured":"Higgins, J.R.: H. Hamburger\u2019s collection of five equivalent results, and some later developments, Technical Report, p. 31 (2007)","DOI":"10.1016\/j.lcats.2007.07.003"},{"issue":"1","key":"57_CR45","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF03549505","volume":"8","author":"JR 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."},{"key":"57_CR46","doi-asserted-by":"crossref","unstructured":"Higgins, J.R.: Paley\u2013Wiener spaces and their reproducing formulae. In: Ruzhansky, M., Wirth, J. (eds.) Progress in Analysis and Its Applications (Proceedings of 7th Internat. ISAAC Congress, London, Jul. 13\u201318, 2009, pp. 273\u2013279. World Scientific Publisher, Hackensack (2010). doi: 10.1142\/9789814313179_0036","DOI":"10.1142\/9789814313179_0036"},{"issue":"1","key":"57_CR47","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/BF03549466","volume":"6","author":"YM Hong","year":"2007","unstructured":"Hong, Y.M., Kim, J.M., Kwon, K.H.: Sampling theory in abstract reproducing kernel Hilbert space. Sampl. Theory Signal Image Process. 6(1), 109\u2013121 (2007)","journal-title":"Sampl. Theory Signal Image Process."},{"key":"57_CR48","doi-asserted-by":"crossref","unstructured":"Ismail, M.E.H., Nashed, M.Z., Zayed, A.I., Ghaleb, A.F., eds. Mathematical Analysis, Wavelets, and Signal Processing, Proceedings of Internat. Conference on Mathematical Analysis and Signal Processing, Cairo, Jan. 3\u20139, American Mathematical Society, Providence, 1995, (1994). doi: 10.1090\/conm\/190","DOI":"10.1090\/conm\/190"},{"key":"57_CR49","volume-title":"The Riemann Zeta-Function","author":"A Ivi\u0107","year":"1985","unstructured":"Ivi\u0107, A.: The Riemann Zeta-Function. Wiley, New York (1985)"},{"issue":"75","key":"57_CR50","doi-asserted-by":"crossref","first-page":"57","DOI":"10.24033\/asens.1066","volume":"3","author":"JP Kahane","year":"1958","unstructured":"Kahane, J.P., Mandelbrojt, S.: Sur l\u2019\u00e9quation fonctionnelle de Riemann et la formule sommatoire de Poisson. Ann. Sci. \u00c9cole Norm. Sup. 3(75), 57\u201380 (1958)","journal-title":"Ann. Sci. \u00c9cole Norm. Sup."},{"key":"57_CR51","doi-asserted-by":"crossref","unstructured":"Karatsuba, A.A., Voronin, S.M.: The Riemann Zeta-Function. Walter de Gruyter & Co., Berlin (1992). doi: 10.1515\/9783110886146","DOI":"10.1515\/9783110886146"},{"key":"57_CR52","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1002\/mana.19911540111","volume":"154","author":"D Klusch","year":"1991","unstructured":"Klusch, D.: The sampling theorem, Dirichlet series and Bessel functions. Math. Nachr. 154, 129\u2013139 (1991). doi: 10.1002\/mana.19911540111","journal-title":"Math. Nachr."},{"issue":"3","key":"57_CR53","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1016\/0377-0427(92)90001-E","volume":"44","author":"D Klusch","year":"1992","unstructured":"Klusch, D.: The sampling theorem, Dirichlet series and Hankel transforms. J. Comput. Appl. Math. 44(3), 261\u2013273 (1992). doi: 10.1016\/0377-0427(92)90001-E","journal-title":"J. Comput. Appl. Math."},{"key":"57_CR54","unstructured":"Kluv\u00e1nek, I.: Sampling theorem in abstract harmonic analysis. Mat.-Fyz. \u010casopis Sloven. Akad. Vied 15, 43\u201348 (1965)"},{"issue":"7","key":"57_CR55","doi-asserted-by":"crossref","first-page":"1915","DOI":"10.1090\/S0002-9939-01-06033-6","volume":"129","author":"M Knopp","year":"2001","unstructured":"Knopp, M., Robins, S.: Easy proofs of Riemann\u2019s functional equation for $$\\zeta (s)$$ \u03b6 ( s ) and of Lipschitz summation. Proc. Am. Math. Soc. 129(7), 1915\u20131922 (2001). doi: 10.1090\/S0002-9939-01-06033-6","journal-title":"Proc. Am. Math. Soc."},{"key":"57_CR56","volume-title":"Lehrbuch der Thetafunktionen","author":"A Krazer","year":"1970","unstructured":"Krazer, A.: Lehrbuch der Thetafunktionen. Chelsea Publishing Co., New York (1970)"},{"key":"57_CR57","doi-asserted-by":"crossref","unstructured":"Mordell, L.J.: Poisson\u2019s summation formula and the Riemann zeta function. J. Lond. Math. Soc.S1\u20134(4), 285 (1929). doi: 10.1112\/jlms\/s1-4.4.285","DOI":"10.1112\/jlms\/s1-4.4.285"},{"key":"57_CR58","volume-title":"Problems in Analytic Number Theory","author":"MR Murty","year":"2008","unstructured":"Murty, M.R.: Problems in Analytic Number Theory, 2nd edn. Springer, New York (2008)","edition":"2"},{"key":"57_CR59","unstructured":"Nashed, M.Z.: Inverse problems, moment problems, signal processing: un menage a trois. In: Siddiqi, A.H., Singh, R.C., Manchanda, P. (eds.) Mathematics in Science and Technology (Proceedings of Satellite Conference of the Internat. Congress of Mathematicians, New Delhi, India, Aug. 14\u201317, 2010). World Scientific Publishers, Hackensack, pp. 2\u201319 (2011)"},{"issue":"7","key":"57_CR60","doi-asserted-by":"crossref","first-page":"2422","DOI":"10.1016\/j.jfa.2009.12.012","volume":"258","author":"MZ Nashed","year":"2010","unstructured":"Nashed, M.Z., Sun, Q.: Sampling and reconstruction of signals in a reproducing kernel subspace of $$L^{p}({\\mathbb{R}}^{d})$$ L p ( R d ) . J. Funct. Anal. 258(7), 2422\u20132452 (2010). doi: 10.1016\/j.jfa.2009.12.012","journal-title":"J. Funct. Anal."},{"issue":"6","key":"57_CR61","doi-asserted-by":"crossref","first-page":"1995","DOI":"10.1090\/S0002-9939-2012-11644-2","volume":"141","author":"MZ Nashed","year":"2013","unstructured":"Nashed, M.Z., Sun, Q., Xian, J.: Convolution sampling and reconstruction of signals in a reproducing kernel subspace. Proc. Am. Math. Soc. 141(6), 1995\u20132007 (2013). doi: 10.1090\/S0002-9939-2012-11644-2","journal-title":"Proc. Am. Math. Soc."},{"key":"57_CR62","doi-asserted-by":"crossref","unstructured":"Nashed, M.Z., Walter, G.G.: Reproducing kernel Hilbert spaces from sampling expansions. In: Ismail, M.E.H., Nashed, M.Z., Zayed, A.I., Ghaleb, A.F. (eds.) Mathematical Analysis, Wavelets, and Signal Processing, pp. 221\u2013226 (1994). doi: 10.1090\/conm\/190\/02304","DOI":"10.1090\/conm\/190\/02304"},{"key":"57_CR63","volume-title":"Analytic Number Theory","author":"DJ Newman","year":"1998","unstructured":"Newman, D.J.: Analytic Number Theory. Springer, New York (1998)"},{"key":"57_CR64","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-65711-5","volume-title":"Approximation of Functions of Several Variables and Imbedding Theorems","author":"SM Nikol\u2019ski\u012d","year":"1975","unstructured":"Nikol\u2019ski\u012d, S.M.: Approximation of Functions of Several Variables and Imbedding Theorems. Springer, New York (1975)"},{"key":"57_CR65","unstructured":"NuHAG (Numerical Harmonic Analysis Group). http:\/\/www.univie.ac.at\/nuhag-php\/home"},{"key":"57_CR66","doi-asserted-by":"crossref","unstructured":"Patterson, S.J.: An Introduction to the Theory of the Riemann Zeta-Function. Cambridge University Press, Cambridge (1988). doi: 10.1017\/CBO9780511623707","DOI":"10.1017\/CBO9780511623707"},{"key":"57_CR67","unstructured":"Pesenson, I.: Best approximations in a space of the representation of a Lie group (russian). Dokl. Akad. Nauk SSSR 302(5), 1055\u20131058 (1988). Trans. Soviet Math. Dokl. 38(2), 384\u2013388 (1989)"},{"issue":"3","key":"57_CR68","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1007\/BF02476027","volume":"4","author":"I Pesenson","year":"1998","unstructured":"Pesenson, I.: Sampling of Paley\u2013Wiener functions on stratified groups. J. Fourier Anal. Appl. 4(3), 271\u2013281 (1998). doi: 10.1007\/BF02476027","journal-title":"J. Fourier Anal. Appl."},{"key":"57_CR69","doi-asserted-by":"crossref","unstructured":"Pesenson, I.: A sampling theorem on homogeneous manifolds. Trans. Am. Math. Soc. 352(9), 4257\u20134269 (2000). doi: 10.1090\/S0002-9947-00-02592-7","DOI":"10.1090\/S0002-9947-00-02592-7"},{"issue":"1","key":"57_CR70","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/s00041-001-0007-9","volume":"7","author":"I Pesenson","year":"2001","unstructured":"Pesenson, I.: Sampling of band-limited vectors. J. Fourier Anal. Appl. 7(1), 93\u2013100 (2001). doi: 10.1007\/s00041-001-0007-9","journal-title":"J. Fourier Anal. Appl."},{"issue":"2","key":"57_CR71","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1016\/j.jat.2005.04.007","volume":"135","author":"I Pesenson","year":"2005","unstructured":"Pesenson, I.: Polynomial splines and eigenvalue approximations on quantum graphs. J. Approx. Theory 135(2), 203\u2013220 (2005). doi: 10.1016\/j.jat.2005.04.007","journal-title":"J. Approx. Theory"},{"issue":"10","key":"57_CR72","doi-asserted-by":"crossref","first-page":"5603","DOI":"10.1090\/S0002-9947-08-04511-X","volume":"360","author":"I Pesenson","year":"2008","unstructured":"Pesenson, I.: Sampling in Paley\u2013Wiener spaces on combinatorial graphs. Trans. Am. Math. Soc. 360(10), 5603\u20135627 (2008). doi: 10.1090\/S0002-9947-08-04511-X","journal-title":"Trans. Am. Math. Soc."},{"key":"57_CR73","volume-title":"Classical Harmonic Analysis and Locally Compact Groups","author":"H Reiter","year":"1968","unstructured":"Reiter, H.: Classical Harmonic Analysis and Locally Compact Groups. Clarendon Press, Oxford (1968)"},{"key":"57_CR74","unstructured":"Riemann, B.: \u00dcber die Anzahl der Primzahlen unter einer gegebenen Gr\u00f6\u00dfe. Monatsber. K\u00f6nigl. Preuss. Akad. Wiss. Berlin Nov. 1859, 671\u2013680. Reprinted. In: Riemann, B., Narasimhan, R. (eds.) Gesammelte mathematische Werke, wissenschaftlicher Nachlass und Nachtr\u00e4ge, (Based on the edition by Weber, H., Dedekind, R.). Springer, Berlin (1990)"},{"issue":"1","key":"57_CR75","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1006\/jmaa.1994.1244","volume":"185","author":"PG Rooney","year":"1994","unstructured":"Rooney, P.G.: Another proof of the functional equation for the Riemann zeta function. J. Math. Anal. Appl. 185(1), 223\u2013228 (1994). doi: 10.1006\/jmaa.1994.1244","journal-title":"J. Math. Anal. Appl."},{"key":"57_CR76","volume-title":"Fourier Analysis on Groups","author":"W Rudin","year":"1962","unstructured":"Rudin, W.: Fourier Analysis on Groups. Interscience Publishers, New York (1962)"},{"key":"57_CR77","volume-title":"Integral Transforms. Reproducing Kernels and Their Applications","author":"S Saitoh","year":"1997","unstructured":"Saitoh, S.: Integral Transforms. Reproducing Kernels and Their Applications. Longman, Harlow (1997)"},{"issue":"1\u20132","key":"57_CR78","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1007\/s00010-004-2743-9","volume":"70","author":"W Schuster","year":"2005","unstructured":"Schuster, W.: Ein kurzer Beweis der Funktionalgleichung der Riemannschen Zetafunktion (a short proof of the functional equation of the riemann zeta function). Aequationes Math. 70(1\u20132), 191\u2013194 (2005). doi: 10.1007\/s00010-004-2743-9","journal-title":"Aequationes Math."},{"issue":"3\u20134","key":"57_CR79","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1007\/BF01457989","volume":"86","author":"C Siegel","year":"1922","unstructured":"Siegel, C.: Bemerkung zu einem Satz von Hamburger \u00fcber die Funktionalgleichung der Riemannschen Zetafunktion. Math. Ann. 86(3\u20134), 276\u2013279 (1922). doi: 10.1007\/BF01457989","journal-title":"Math. Ann."},{"key":"57_CR80","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-2706-9","volume-title":"Numerical Methods Based on sinc and Analytic Functions","author":"F Stenger","year":"1993","unstructured":"Stenger, F.: Numerical Methods Based on sinc and Analytic Functions. Springer, New York (1993)"},{"key":"57_CR81","doi-asserted-by":"crossref","unstructured":"Stenger, F.: Sinc convolution-a tool for circumventing some limitations of classical signal processing. In: Ismail, M.E.H., Nashed, M.Z., Zayed, A.I., Ghaleb, A.F. (eds.) Mathematical Analysis, Wavelets, and Signal Processing, pp. 227\u2013240 (1994). doi: 10.1090\/conm\/190\/02305","DOI":"10.1090\/conm\/190\/02305"},{"key":"57_CR82","volume-title":"The Theory of the Riemann Zeta-Function","author":"EC Titchmarsh","year":"1986","unstructured":"Titchmarsh, E.C.: The Theory of the Riemann Zeta-Function, 2nd edn. Clarendon Press, New York (1986)","edition":"2"},{"key":"57_CR83","unstructured":"Weiss, P.: An estimate of the error arising from misapplication of the sampling theorem. Notices Am. Math. Soc. 10, 351 (1963). Abstract No. 601\u201354"},{"key":"57_CR84","volume-title":"Advances in Shannon\u2019s Sampling Theory","author":"AI Zayed","year":"1993","unstructured":"Zayed, A.I.: Advances in Shannon\u2019s Sampling Theory. CRC Press, Boca Raton (1993)"},{"key":"57_CR85","unstructured":"Zygmund, A.: Trigonometric Series. vols. I, II, 2nd edn., reprinted with corrections and some additions (two volumes bound as one) edn. Cambridge University Press, London (1968)"}],"container-title":["Bulletin of Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s13373-014-0057-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s13373-014-0057-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s13373-014-0057-3","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,5]],"date-time":"2025-05-05T04:50:45Z","timestamp":1746420645000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s13373-014-0057-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,10,17]]},"references-count":85,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2014,12]]}},"alternative-id":["57"],"URL":"https:\/\/doi.org\/10.1007\/s13373-014-0057-3","relation":{},"ISSN":["1664-3607","1664-3615"],"issn-type":[{"type":"print","value":"1664-3607"},{"type":"electronic","value":"1664-3615"}],"subject":[],"published":{"date-parts":[[2014,10,17]]}}}