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Transactions of the American Mathematical Society

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Convolution, differential equations, and entire functions of exponential type

Author: Dale H. Mugler
Journal: Trans. Amer. Math. Soc. 216 (1976), 145-187
MSC: Primary 30A64; Secondary 34A20
MathSciNet review: 0387587
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Abstract: The Whittaker-Shannon interpolation formula, or ``cardinal series", is a special case of the more general linear integro-differential equation with constant complex coefficients $ \Sigma _{n = 0}^\infty {a_n}{f^{(n)}}(z) = \smallint f(z - t)d\mu (t)$ where the integral is taken over the whole real line with respect to the measure $ \mu $.

In this study, I show that many of these equations provide representations for particular classes of entire functions of exponential type. That is, every function in the class satisfies the equation and conversely every solution of the equation is a member of the class of functions.

When the measure in the convolution integral above is chosen to be discrete, a particular form of the above type of equation is an equation of periodicity $ f(z) = f(z + \tau )$. Following an extensive treatment of the general equation written above, the study concludes by offering a generalization in terms of these convolution equations of a classical theorem in complex analysis concerning periodic entire functions.

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  • [1] Ralph Philip Boas Jr., Entire functions, Academic Press Inc., New York, 1954. MR 0068627
  • [2] R. P. Boas Jr., Functions of exponential type. III, Duke Math. J. 11 (1944), 507–511. MR 0010723
  • [3] R. P. Boas Jr. and H. Pollard, Continuous analogues of series, Amer. Math. Monthly 80 (1973), 18–25. MR 0315354
  • [4] Louis de Branges, Hilbert spaces of entire functions, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1968. MR 0229011
  • [5] Avner Friedman, Generalized functions and partial differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. MR 0165388
  • [6] A. O. Gel′fond, Linear differential equations of infinite order with constant coefficients and asymptotic periods of entire functions, Trudy Mat. Inst. Steklov., v. 38, Izdat. Akad. Nauk SSSR, Moscow, 1951, pp. 42–67 (Russian). MR 0047776
  • [7] André Giroux, Une remarque sur l'interpolation des fonctions entières, Private communication, 1973.
  • [8] I. Halperin and H. R. Pitt, Integral inequalities connected with differential operators, Duke Math. J. 4 (1938), no. 3, 613–625. MR 1546080, 10.1215/S0012-7094-38-00451-X
  • [9] G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. MR 0030620
  • [10] David Jagerman, Information theory and approximation of bandlimited functions, Bell System Tech. J. 49 (1970), 1911–1941. MR 0320600
  • [11] A. Kolmogorov, On inequalities between the upper bounds of successive derivatives of an arbitrary function on an infinite interval, Učen. Zap. Moskov. Gos. Univ. Matematika 30 (1939), 3-16; English transl., Amer. Math. Soc. Transl. (1) 2 (1962), 233-243. MR 1, 298, 400; 11, 86.
  • [12] Henry P. Kramer, The digital form of operators on band-limited functions, J. Math. Anal. Appl. 44 (1973), 275–287. MR 0329744
  • [13] Edmund Landau, Über einen Satz von Herrn Esclangon, Math. Ann. 102 (1930), no. 1, 177–188 (German). MR 1512573, 10.1007/BF01782342
  • [14] A. J. Macintyre, Laplace's transformation and integral functions, Proc. London Math. Soc. (2) 45 (1938), 1-20.
  • [15] Thomas J. Osler, A further extension of the Leibniz rule to fractional derivatives and its relation to Parseval’s formula, SIAM J. Math. Anal. 3 (1972), 1–16. MR 0323970
  • [16] G. Pólya, Sur certaines transformations fonctionnelles lineaires des fonctions analytiques, Math. Z. 29 (1929), 549-640.
  • [17] H. R. Pitt, On the class of integro-differential equations, Proc. Cambridge Philos. Soc. 40 (1944), 199–211. MR 0012197
  • [18] Walter Rudin, Functional analysis, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR 0365062
  • [19] I. J. Schoenberg, The elementary cases of Landau’s problem of inequalities between derivatives, Amer. Math. Monthly 80 (1973), 121–158. MR 0315070
  • [20] P. C. Sikkema, Differential operators and differential equations of infinite order with constant coefficients. Researches in connection with integral functions of finite order, P. Noordhoff N. V., Groningen-Djakarta, 1953. MR 0060082
  • [21] J. Tagamlizki, Funktionen, die auf der reellen achse gewissen Ungleichungen genügen, Annuaire [Godišnik] Univ. Sofia. Fac. Phys.-Math. Livre 1. 42 (1946), 239–256 (Bulgarian, with German summary). MR 0021042
  • [22] J. M. Whittaker, Interpolator function theory, Cambridge Univ. Press, London, 1935.

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Keywords: Entire function, exponential type, interpolation, integro-differential equation, differential-difference equation, convolution, Laplace transform, distribution
Article copyright: © Copyright 1976 American Mathematical Society