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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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Five short stories about the cardinal series
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by J. R. Higgins PDF
Bull. Amer. Math. Soc. 12 (1985), 45-89
References
    A. V. Balakrishnan (1957), A note on the sampling principle for continuous signals, IRE Trans. Inform. Theory IT-3, 143-146.
  • A. B. Bhatia and K. S. Krishnan, Light-scattering in homogeneous media regarded as reflexion from appropriate thermal elastic waves, Proc. Roy. Soc. London Ser. A 192 (1948), 181–194. MR 25612, DOI 10.1098/rspa.1948.0004
    • C. Bigelow and D. Day (1983), Digital typography, Scientific American (2) 249, 94-105. H. S. Black (1953), Modulation theory, van Nostrand, Princeton, N. J. V. Blažek (1974), Sampling theorem and the number of degrees of freedom of an image, Optics Comm. 11, 144-147. V. Blažek (1976), Optical information processing by the Fabry-Perot resonator, Optical Quantum Electronics 8, 237-240.
  • Ralph Philip Boas Jr., Entire functions, Academic Press, Inc., New York, 1954. MR 0068627
  • R. P. Boas Jr., Summation formulas and band-limited signals, Tohoku Math. J. (2) 24 (1972), 121–125. MR 330915, DOI 10.2748/tmj/1178241524
  • R. P. Boas Jr. and H. Pollard, Continuous analogues of series, Amer. Math. Monthly 80 (1973), 18–25. MR 315354, DOI 10.2307/2319254
    • F. E. Bond and C. R. Cahn (1958), On sampling the zeros of bandwidth limited signals, IRE Trans. Inform. Theory IT-4, 110-113. E. Borel (1897), Sur l’interpolation, C. R. Acad. Sci. Paris 124, 673-676. E. Borel (1898), Sur la recherche des singularités d’une fonction définie par un développement de Taylor, C. R. Acad. Sci. Paris 127, 1001-1003.
  • Emile Borel, Mémoire sur les séries divergentes, Ann. Sci. École Norm. Sup. (3) 16 (1899), 9–131 (French). MR 1508965, DOI 10.24033/asens.463
  • 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 (1967), 75–84. MR 204952, DOI 10.1016/0022-247X(67)90183-7
    • J. L. Brown, Jr. (1980), First order sampling of bandpass signals—a new approach, IEEE Trans. Inform. Theory IT-26, 613-615.
  • John L. Brown Jr., Multichannel sampling of low-pass signals, IEEE Trans. Circuits and Systems 28 (1981), no. 2, 101–106. MR 600953, DOI 10.1109/TCS.1981.1084954
    • T. A. Brown (1915-1916), Fourier’s integral, Proc. Edinburgh Math. Soc. 34, 3-10. H. Burkhardt (1899-1916), Trigonometrische Interpolation, Enzyklopädie Math. Wiss. IIA9a, Teubner, Leipzig.
  • P. L. Butzer, A survey of the Whittaker-Shannon sampling theorem and some of its extensions, J. Math. Res. Exposition 3 (1983), no. 1, 185–212. MR 724869
  • P. L. Butzer and W. Engels, Dyadic calculus and sampling theorems for functions with multidimensional domain. I. General theory, Inform. and Control 52 (1982), no. 3, 333–351. MR 707580, DOI 10.1016/S0019-9958(82)90806-3
  • Paul L. Butzer and Rolf J. Nessel, Fourier analysis and approximation, Pure and Applied Mathematics, Vol. 40, Academic Press, New York-London, 1971. Volume 1: One-dimensional theory. MR 0510857, DOI 10.1007/978-3-0348-7448-9
    • P. L. Butzer and W. Splettstösser (1977), Approximation und interpolation durch verallgemeinerte Abtastsummen, Westdeutscher Verlag, Opladen.
  • P. L. Butzer and R. L. Stens, The Euler-MacLaurin summation formula, the sampling theorem, and approximate integration over the real axis, Linear Algebra Appl. 52/53 (1983), 141–155. MR 709348, DOI 10.1016/0024-3795(83)80011-1
  • L. L. Campbell, Sampling theorem for the Fourier transform of a distribution with bounded support, SIAM J. Appl. Math. 16 (1968), 626–636. MR 227692, DOI 10.1137/0116051
    • M. L. Cartwright (1936), On certain integral functions of order one, Quart. J. Math. Oxford Ser. 7, 46-55. A. L. Cauchy (1841), Mémoire sur diverses formules d’analyse, C. R. Acad. Sci. Paris 12, 283-298. P. Cazzaniga (1882), Esspressione di funzioni intere che in posti dati arbitrariamente prendono valori prestabiliti, Ann. Mat. Pura Appl. (2) 10, 278-290.
  • D. K. Cheng and D. L. Johnson, Walsh transform of sampled time functions and the sampling principle, Proc. IEEE 61 (1973), 674–675. MR 0345697, DOI 10.1109/PROC.1973.9133
  • Ronald R. Coifman and Guido Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645. MR 447954, DOI 10.1090/S0002-9904-1977-14325-5
    • W. L. Ferrar (1925), On the cardinal function of interpolation-theory, Proc. Roy. Soc. Edinburgh 45, 267-282. W. L. Ferrar (1926), On the cardinal function of interpolation-theory, Proc. Roy. Soc. Edinburgh 46, 323-333. L. J. Fogel (1955), A note on the sampling theorem, IRE Trans. Inform. Theory 1, 47-48. C. F. Gauss (1900), Carl Friedrich Gauss Werke, Band 8, Königl. Gesellschaft Wiss. Gottingen, Teubner, Leipzig.
  • Stanford Goldman, Information theory, Prentice-Hall, Inc., New York, 1953. MR 0064349
    • E. A. Gonzáles-Velasco and E. Sanvicente (1980), The analytic representation of bandpass signals, J. Franklin Inst. 310, 135-142.
  • R. P. Gosselin, On the $L^{p}$ theory of cardinal series, Ann. of Math. (2) 78 (1963), 567–581. MR 156154, DOI 10.2307/1970542
  • R. P. Gosselin, Singular integrals and cardinal series, Studia Math. 44 (1972), 39–45. MR 320824, DOI 10.4064/sm-44-1-39-45
    • M. Guichard (1884), Sur les fonctions entières, Ann, Sci. École Norm. Sup. (3) 1, 427-432. J. Hadamard (1901), La série de Taylor et son prolongement analytique, Scientia 12, 1-100. A. H. Haddad, K. Yao and J. B. Thomas (1967), General methods for the derivation of sampling theorems, IEEE Trans. Inform. Theory IT-13, 227-230.
  • G. H. Hardy, Notes on special systems of orthogonal functions. IV. The orthogonal functions of Whittaker’s cardinal series, Proc. Cambridge Philos. Soc. 37 (1941), 331–348. MR 5145, DOI 10.1017/s0305004100017977
  • J. R. Higgins, An interpolation series associated with the Bessel-Hankel transform, J. London Math. Soc. (2) 5 (1972), 707–714. MR 320616, DOI 10.1112/jlms/s2-5.4.707
  • J. R. Higgins, A sampling theorem for irregularly spaced sample points, IEEE Trans. Inform. Theory IT-22 (1976), no. 5, 621–622. MR 416736, DOI 10.1109/tit.1976.1055596
  • John Rowland Higgins, Completeness and basis properties of sets of special functions, Cambridge University Press, Cambridge-New York-Melbourne, 1977. Cambridge Tracts in Mathematics, No. 72. MR 0499341, DOI 10.1017/CBO9780511566189
    • I. I. Hirschman (1964), Review of "On the L, Math. Rev. 27 # 6086, 1163. D. L. Jagerman and L. J. Fogel (1956), Some general aspects of the sampling theorem, IEEE Trans. Inform. Theory 2, 139-156. A. J. Jerri (1977), The Shannon sampling theorem—its various extensions and applications: a tutorial review, Proc. IEEE 65, 1565-1596. P. E. B. Jourdain (1905), On the general theory of functions, J. Reine Angew. Math. 128, 169-210. S. C. Kak (1970), Sampling theorem in Walsh-Fourier analysis, Electronics Lett. 6, 447-448.
  • Masahiko Kawamura and Sueo Tanaka, Proof of sampling theorem in sequency analysis using extended Walsh functions, Systems-Comput.-Controls 9 (1978), no. 5, 10–15 (1980). MR 589857
    • Y. I. Khurgin and V. P. Yakovlev (1977), Progress in the Soviet Union on the theory and applications of bandlimited functions, Proc. IEEE 65, 1005-1029.
  • Igor Kluvánek, Sampling theorem in abstract harmonic analysis, Mat.-Fyz. Časopis. Sloven. Akad. Vied. 15 (1965), 43–48 (English, with Russian summary). MR 188717
  • Igor Kluvánek, Positive-definite signals with maximal energy, J. Math. Anal. Appl. 39 (1972), 580–585. MR 313719, DOI 10.1016/0022-247X(72)90182-5
  • Arthur Kohlenberg, Exact interpolation of band-limited functions, J. Appl. Phys. 24 (1953), 1432–1436. MR 60630, DOI 10.1063/1.1721195
    • A. N. Kolmogorov (1956), On the Shannon theory of information transmission in the case of continuous signals, IRE Trans. Inform. Theory IT-2, 102-108. A. N. Kolmogorov and V. M. Tihomirov (1960), ε-Entropie und ε-Kapazität von Mengen in Funktional Räumen, Math. Forschungsberichte 10, VEB Deutscher Verlag Wiss., Berlin. (Translated from the Russian) V. A. Kotel’nikov (1933), On the carrying capacity of the "ether" and wire in telecommunications, Material for the First All-Union Conference on Questions of Communication, Izd. Red. Upr. Svyazi RKKA, Moscow. (Russian)
  • H. P. Kramer, A generalized sampling theorem, J. Math. and Phys. 38 (1959/60), 68–72. MR 103786, DOI 10.1002/sapm195938168
  • Henry P. Kramer, The digital form of operators on band-limited functions, J. Math. Anal. Appl. 44 (1973), 275–287. MR 329744, DOI 10.1016/0022-247X(73)90059-0
    • H. J. Landau (1967a), Sampling, data transmission, and the Nyquist rate, Proc. IEEE 55, 1701-1706.
  • H. J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967), 37–52. MR 222554, DOI 10.1007/BF02395039
  • H. J. Landau and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty. III. The dimension of the space of essentially time- and band-limited signals, Bell System Tech. J. 41 (1962), 1295–1336. MR 147686, DOI 10.1002/j.1538-7305.1962.tb03279.x
    • D. A. Linden (1959), A discussion of sampling theorems, Proc. IRE 47, 1219-1226.
  • D. A. Linden and N. M. Abramson, A generalization of the sampling theorem, Information and Control 3 (1960), 26–31. MR 110592, DOI 10.1016/S0019-9958(60)90242-4
    • H. D. Lüke (1978), Zur Entstehung des Abtasttheorems, Nachr. Techn. Z. 31, 271-274. A. J. Macintyre (1938), Laplace’s transformation and integral functions, Proc. London Math. Soc. 45, 1-20.
  • Wilhelm Magnus, Fritz Oberhettinger, and Raj Pal Soni, Formulas and theorems for the special functions of mathematical physics, Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag New York, Inc., New York, 1966. MR 0232968, DOI 10.1007/978-3-662-11761-3
    • M. Maqusi (1972), Walsh functions and the sampling principle, Proc. Walsh Functions Sympos., U. S. Naval Research Lab. E. Masry (1982), The application of random reference sequences to the reconstruction of clipped differentiable signals, IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 953-963.
  • F. C. Mehta, Sampling expansion for band-limited signals through some special functions, J. Cybernet. 5 (1975), no. 2, 61–68 (1976). MR 416739, DOI 10.1080/01969727508546090
    • R. M. Mersereau (1979), The processing of hexagonally sampled two dimensional signals, Proc. IEEE 67, 930-949. R. M. Mersereau and T. C. Speake (1983), The processing of periodically sampled multidimensional signals, IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 188-194.
  • Herbert Meschkowski, Hilbertsche Räume mit Kernfunktion, Die Grundlehren der mathematischen Wissenschaften, Band 113, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1962 (German). MR 0140912, DOI 10.1007/978-3-642-94848-0
  • Dale H. Mugler, Convolution, differential equations, and entire functions of exponential type, Trans. Amer. Math. Soc. 216 (1976), 145–197. MR 387587, DOI 10.1090/S0002-9947-1976-0387587-3
  • Dale H. Mugler, The discrete Paley-Wiener theorem, J. Math. Anal. Appl. 75 (1980), no. 1, 172–179. MR 576282, DOI 10.1016/0022-247X(80)90314-5
  • Jacques Neveu, Le problème de l’échantillonnage et de l’interpolation d’un signal, C. R. Acad. Sci. Paris 260 (1965), 49–51 (French). MR 172732
    • K. Ogura (1920), On some central difference formulas of interpolation, Tôhoku Math. J. 17, 232-241.
  • Fritz Oberhettinger, Tabellen zur Fourier Transformation, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957 (German). MR 0081997, DOI 10.1007/978-3-642-94700-1
    • A. Papoulis (1968), Systems and transforms with applications in optics, McGraw-Hill, New York. E. Parzen (1956), A simple proof and some extensions of sampling theorems, Tech. Rep. 7, Stanford Univ., Stanford. D. P. Petersen (1963), Sampling of space-time stochastic processes with application to information and design systems, Thesis, Rensselaer Polytechnic Inst., Troy, N. Y.
  • Daniel P. Petersen and David Middleton, Sampling and reconstruction of wave-number-limited functions in $N$-dimensional Euclidean spaces, Information and Control 5 (1962), 279–323. MR 151331, DOI 10.1016/S0019-9958(62)90633-2
    • F. Pichler (1973), Walsh functions—introduction to the theory, Signal Processing (Proc. NATO Advanced Study Inst. Signal Processing, J. W. R. Griffiths et al., eds.), Academic Press, London and New York, pp. 23-41. S.-D. Poisson (1820), Mémoire sur la manière d’exprimer les fonctions, par des séries de quantités périodiques, et sur l’usage de cette transformation dans la résolution de différens problèmes, J. École Roy. Polytechnique 11, 417-489. G. Pólya (1931), Aufgabe 105, Jahresber. Deutsch. Math.-Verein. 40, 80.
  • Reese T. Prosser, A multidimensional sampling theorem, J. Math. Anal. Appl. 16 (1966), 574–584. MR 202496, DOI 10.1016/0022-247X(66)90163-6
  • C. Ryavec, A nonlinear analogue of the cardinal series, Quart. J. Math. Oxford Ser. (2) 30 (1979), no. 118, 223–227. MR 534834, DOI 10.1093/qmath/30.2.223
  • C. E. Shannon, A mathematical theory of communication, Bell System Tech. J. 27 (1948), 379–423, 623–656. MR 26286, DOI 10.1002/j.1538-7305.1948.tb01338.x
  • Claude E. Shannon, Communication in the presence of noise, Proc. I.R.E. 37 (1949), 10–21. MR 28549, DOI 10.1109/JRPROC.1949.232969
    • I. Someya (1949), Waveform transmission, Shukyo, Tokyo.
  • B. Spain, Interpolated derivatives, Proc. Roy. Soc. Edinburgh 60 (1940), 134–140. MR 1779, DOI 10.1017/S0370164600020125
  • B. Spain, Interpolated derivatives, Proc. Edinburgh Math. Soc. (2) 9 (1958), 166–167. MR 114098, DOI 10.1017/S0013091500014061
    • W. Splettstösser (1980), Error analysis in the Walsh sampling theorem, IEEE Sympos. Electromagnetic Computibility, Baltimore. Inst, for Electrical and Electronics Engineers, Service Center, Piscataway, N. J., pp. 366-370.
  • Wolfgang Splettstösser, Sampling approximation of continuous functions with multidimensional domain, IEEE Trans. Inform. Theory 28 (1982), no. 5, 809–814. MR 680150, DOI 10.1109/TIT.1982.1056561
  • W. Splettstösser, R. L. Stens, and G. Wilmes, On approximation by the interpolating series of G. Valiron, Funct. Approx. Comment. Math. 11 (1981), 39–56. MR 692712
  • Henry Stark, Sampling theorems in polar coordinates, J. Opt. Soc. Amer. 69 (1979), no. 11, 1519–1525. MR 550983, DOI 10.1364/JOSA.69.001519
    • H. Stark and C. S. Sarna (1979), Image reconstruction using polar sampling theorems, Appl. Optics 18, 2086-2088.
  • J. F. Steffensen, Über eine Klasse von Ganzen Funktionen und Ihre Anwendung auf die Zahlentheorie, Acta Math. 37 (1914), no. 1, 75–112 (German). MR 1555095, DOI 10.1007/BF02401830
  • R. L. Stens, Error estimates for sampling sums based on convolution integrals, Inform. and Control 45 (1980), no. 1, 37–47. MR 582144, DOI 10.1016/S0019-9958(80)90857-8
    • M. Theis (1919), Uber eine Interpolations—formeln von de la Vallée Poussin, Math. Z. 3, 93-113.
  • E. C. Titchmarsh, Reciprocal formulae involving series and integrals, Math. Z. 25 (1926), no. 1, 321–347. MR 1544814, DOI 10.1007/BF01283842
    • E. C. Titchmarsh (1926b), The zeros of certain integral functions, Proc. London Math. Soc. 25, 283-302. E. C. Titchmarsh (1948), Introduction to the theory of Fourier integrals, 2nd ed., Clarendon Press, Oxford. L. Tschakaloff (1933), Zweite Losung der Aufgabe 105, Jahresber. Deutsch. Math.-Verein. 43, 11-12. G. Valiron (1925), Sur la formule d’interpolation de Lagrange, Bull. Sci. Math. (2) 49, 181-192; 203-224. Ch.-J. de la Vallée Poussin (1908), Sur la convergence des formules d’interpolation entre ordonnées equidistantes, Acad. Roy. Belg. Bull. Cl. Sci. 1, 319-410.
  • J. D. Weston, The cardinal series in Hilbert space, Proc. Cambridge Philos. Soc. 45 (1949), 335–341. MR 30026, DOI 10.1017/S0305004100024944
  • J. D. Weston, A note on the theory of communication, Philos. Mag. (7) 40 (1949), 449–453. MR 29124, DOI 10.1080/14786444908521732
    • E. T. Whittaker (1915), On the functions which are represented by the expansions of the interpolation theory, Proc. Roy. Soc. Edinburgh 35, 181-194. J. M. Whittaker (1929a), On the cardinal function of interpolation theory, Proc. Edinburgh Math. Soc. 1, 41-46. J. M. Whittaker (1929b), On the "Fourier theory" of the cardinal function, Proc. Edinburgh Math. Soc. 1, 169-176. J. M. Whittaker (1935), Interpolatory function theory, Cambridge Univ. Press, Cambridge. J. L. Yen (1956), On nonuniform sampling of bandwidth-limited signals, IRE Trans. Circuit Theory CT-3, 251-257.
  • Robert M. Young, An introduction to nonharmonic Fourier series, Pure and Applied Mathematics, vol. 93, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 591684
  • Moshe Zakai, Band-limited functions and the sampling theorem, Information and Control 8 (1965), 143–158. MR 174403, DOI 10.1016/S0019-9958(65)90038-0
    • W. Ziegler (1981), Haar-Fourier Transformation auf dem R+, Doctoral Diss. Technische Univ. München.
  • A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
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Additional Information
  • Journal: Bull. Amer. Math. Soc. 12 (1985), 45-89
  • MSC (1980): Primary 41A05, 42C10; Secondary 41-03, 01A55, 42B99, 42C30, 94-03, 01A60, 94A05
  • DOI: https://doi.org/10.1090/S0273-0979-1985-15293-0
  • MathSciNet review: 766960