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Bulletin of the American Mathematical Society

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

 
 

 

Five short stories about the cardinal series


Author: J. R. Higgins
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
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  • 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 (1948), Light-scattering in homogeneous media regarded as reflexion from appropriate thermal elastic wave, Proc. Roy. Soc. London Ser. A 192, 181-194. MR 25612
  • 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.
  • R. P. Boas, Jr. (1954), Entire functions, Academic Press, New York. MR 68627
  • R. P. Boas, Jr. (1972), Summation formulas and band-limited signals, Tôhoku Math. J. 24, 121-125. MR 330915
  • R. P. Boas, Jr. and H. Pollard (1973), Continuous analogs of series, Amer. Math. Monthly 80, 18-25. MR 315354
  • 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
  • J. L. Brown, Jr. (1967), On the error in reconstructing a non-bandlimited function by means of the bandpass sampling theorem, J. Math. Anal. Appl. 18, 75-84; Erratum, ibid. 21 (1968), 699. MR 204952
  • J. L. Brown, Jr. (1980), First order sampling of bandpass signalsa 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, https://doi.org/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, https://doi.org/10.1016/S0019-9958(82)90806-3
  • P. L. Butzer and R. Nessel (1971), Fourier analysis and approximation, Vol. 1, Academic Press, New York and London. MR 510857
  • 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, https://doi.org/10.1016/0024-3795(83)80011-1
  • L. L. Campbell (1968), Sampling theorem for the Fourier transform of a distribution with bounded support, SIAM J. Appl. Math. 16, 626-636. MR 227692
  • 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 (1973), Walsh transform of sampled time functions and the sampling principle, Proc. IEEE 61, 674-675. MR 345697
  • R. R. Coifman and G. Weiss (1977), Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83, 569-645. MR 447954
  • 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.
  • S. Goldman (1953), Information theory, Constable, London. MR 64349
  • E. A. Gonzáles-Velasco and E. Sanvicente (1980), The analytic representation of bandpass signals, J. Franklin Inst. 310, 135-142.
  • R. P. Gosselin (1963), On the L, Ann. of Math. (2) 78, 567-581. MR 156154
  • R. P. Gosselin (1972), Singular integrals and cardinal series, Studia Math. 44, 39-45. MR 320824
  • 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 (1941), Notes on special systems of orthogonal functions, IV: The orthogonal functions of Whittaker's cardinal series, Proc. Cambridge Philos. Soc. 37, 331-348. MR 5145
  • J. R. Higgins (1972), An interpolation series associated with the Bessel-Hankel transform, J. London Math. Soc. 5, 707-714. MR 320616
  • J. R. Higgins (1976), A sampling theorem for irregularly spaced sample points, IEEE Trans. Inform. Theory IT-22, 621-622. MR 416736
  • J. R. Higgins (1977), Completeness and basis properties of sets of special functions, Cambridge Univ. Press, Cambridge. MR 499341
  • 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 theoremits 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.
  • I. Kluvánek (1965), Sampling theorem in abstract harmonic analysis, Mat-Fyz. Časopis Sloven. Akad. Vied 15, 43-48. MR 188717
  • I. Kluvánek (1972), Positive definite signals with maximal energy, J. Math. Anal. Appl. 39, 580-585. MR 313719
  • A. Kohlenberg (1953), Exact interpolation of bandlimited functions, J. Appl. Phys. 24, 1432-1436. MR 60630
  • 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 (1957), A generalised sampling theorem, J. Math. Phys. 63, 68-72. MR 103786
  • H. P. Kramer (1973), The digital form of operators on band-limited functions, J. Math. Anal. Appl. 44, 275-287. MR 329744
  • H. J. Landau (1967a), Sampling, data transmission, and the Nyquist rate, Proc. IEEE 55, 1701-1706.
  • H. J. Landau (1967b), Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117, 37-52. MR 222554
  • H. J. Landau and H. O. Pollack (1962), Prolate spheroidal wave functions, Fourier analysis and uncertainty. III, Bell System Tech. J. 41, 1295-1336. MR 147686
  • D. A. Linden (1959), A discussion of sampling theorems, Proc. IRE 47, 1219-1226.
  • D. A. Linden and N. M. Abramson (1960), A generalisation of the sampling theorem, Inform, and Control 3, 26-31; Errata, Ibid. 4 (1961), 95-96. MR 110592
  • 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.
  • W. Magnus, F. Oberhettinger and R. P. Soni (1966), Formulas and theorems for the special functions of mathematical physics, 3rd ed., Springer-Verlag, Berlin. MR 232968
  • 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 (1975), Sampling expansion for band-limited signals through some special functions, J. Cybernetics 61-68. MR 416739
  • 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.
  • H. Meschowski (1962), Hilbertsche Räume mit kernfunktion, Springer-Verlag, Berlin. MR 140912
  • D. H. Mugler (1976), Convolution, differential equations and entire functions of exponential type, Trans. Amer. Math. Soc. 216, 145-187. MR 387587
  • Dale H. Mugler, The discrete Paley-Wiener theorem, J. Math. Anal. Appl. 75 (1980), no. 1, 172–179. MR 576282, https://doi.org/10.1016/0022-247X(80)90314-5
  • J. Neveu (1965), Le problème d'échantillonnage et de l'interpolation d'un signal, C. R. Acad. Sci. Paris 260, 49-51. MR 172732
  • K. Ogura (1920), On some central difference formulas of interpolation, Tôhoku Math. J. 17, 232-241.
  • F. Oberhettinger (1957), Tabellen zur Fourier Transformation, Springer-Verlag, Berlin. MR 81997
  • 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.
  • D. P. Petersen and D. Middleton (1962), Sampling and reconstruction of wave number-limited functions in N-dimensional Euclidean spaces, Inform, and Control 5, 279-323. MR 151331
  • 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.
  • R. T. Prosser (1966), A multidimensional sampling theorem, J. Math. Anal. Appl. 16, 574-584. MR 202496
  • C. Ryavec, A nonlinear analogue of the cardinal series, Quart. J. Math. Oxford Ser. (2) 30 (1979), no. 118, 223–227. MR 534834, https://doi.org/10.1093/qmath/30.2.223
  • C. E. Shannon (1948), A mathematical theory of communication, Bell System Tech. J. 27, 379-423. MR 26286
  • C. E. Shannon (1949), Communication in the presence of noise, Proc. IRE 37, 10-21. MR 28549
  • I. Someya (1949), Waveform transmission, Shukyo, Tokyo.
  • B. Spain (1940), Interpolated derivatives, Proc. Roy. Soc. Edinburgh 60, 134-140. MR 1779
  • B. Spain (1958), Interpolated derivatives, Proc. Edinburgh Math. Soc. 9, 166-167. MR 114098
  • 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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/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 (1949a), The cardinal series in Hilbert space, Proc. Cambridge Philos. Soc. 45, 335-341. MR 30026
  • J. D. Weston (1949b), A note on the theory of cummunication, Philos. Mag. (303) 40, 449-453. MR 29124
  • 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
  • M. Zakai (1965), Band-limited functions and the sampling theorem, Inform, and Control 8, 143-158. MR 174403
  • W. Ziegler (1981), Haar-Fourier Transformation auf dem R+, Doctoral Diss. Technische Univ. München.
  • A. Zygmund (1959), Trigonometric series, Vol. II. 2nd ed., Cambridge Univ. Press, Cambridge. MR 107776

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DOI: https://doi.org/10.1090/S0273-0979-1985-15293-0

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