Hostname: page-component-7dd5485656-wlg5v Total loading time: 0 Render date: 2025-10-31T21:53:37.508Z Has data issue: false hasContentIssue false
  • English
  • Français

Reconstruction of Entire Functions From Irregularly Spaced Sample Points

Published online by Cambridge University Press:  20 November 2018

Georgi R. Grozev  and
Qazi I. Rahman
Georgi R. Grozev
Affiliation:
Département de Mathématiques et de Statistique, Université de Montréal, Montréal, Québec, H3C 3J7, e-mail: georgig@numetrix.com, e-mail: rahmanqi@ere.umontreal.ca
Qazi I. Rahman
Affiliation:
Département de Mathématiques et de Statistique, Université de Montréal, Montréal, Québec, H3C 3J7, e-mail: georgig@numetrix.com, e-mail: rahmanqi@ere.umontreal.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let where {λn }n ∈ Ζ is a sequence of real numbers such that |λn — n| ≤ Δ for some Δ > 0 and all n ∈ ℤ . Extending an obvious property of sin πz to which the function G reduces when Δ = 0 we show that is bounded by a constant independent of n. The result is then applied to a problem concerning derivative sampling in one and several variables.

Keywords

30D1030D1541A0594A05entire functions of exponential typeWhittaker-Shannon sampling theoremnonuniform samplingmultidimensional samplinginterpolation

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 4 , 01 August 1996 , pp. 777 - 793
Copyright
Copyright © Canadian Mathematical Society 1996

References

1. Ahiezer, N.I., On the interpolation of entire transcendental functions of finite order, Dokl. Akad. Nauk, SSSR(N.S.) 65(1949), 781–784.Google Scholar
2. Boas, R.P., Jr., Integrability along a line for a class of entire functions, Trans. Amer. Math. Soc. 73(1952), 191–197.Google Scholar
3. Boas, R.P., Entire functions, Academic Press, New York, (1954).Google Scholar
4. Butzer, P.L. and Hinsen, G., Two-dimensional nonuniform sampling expansions —An iterative approach. I. Theory of two-dimensional bandlimited signals. II.Reconstruction formulae and applications, Appl. Anal., 32(1989), 53–67. 69-85.Google Scholar
5. Duffin, R.J. and Schaeffer, A.C., class ofnonharmonic Fourier series, Trans. Amer. Math. Soc. 72(1952), 341–366.Google Scholar
6. Fichtenholz, G.M., Differential und Integralrechnung, VEB Deutscher Verlag der Wissenschaften, Berlin 1(1966).Google Scholar
7. Higgins, J.R., A sampling theorem for irregularly spaced sample points, IEEE Trans. Inform. Theory, IT- 22(1976), 621–622.Google Scholar
8. Higgins, J.R., Sampling theorems and the contour integral method, Appl. Anal. 41(1991), 155–168.Google Scholar
9. Hinsen, G., Irregular Sampling of Bandlimited IP —functions, J. Approx., Theory, 72(1993), 346–364.Google Scholar
10. Ya, B.. Levin, On functions of finite degree, bounded on a sequence of points, Dokl. Akad. Nauk SSSR, (N.S.) 65(1949), 265–268.Google Scholar
11. Levinson, N., Gap and density theorems, Amer. Math. Soc. Colloquium Publications, New, York 26(1940).Google Scholar
12. Nikol'skiĭ, S.M., Approximation of functions of several variables and imbedding theorems, Springer-Verlag, Berlin, Heidelberg, New York, 1975.Google Scholar
13. Plancherel, M. and Pólya, G., Fonctions entières et intégrates de Fourier multiples, Comment. Math. Helv. 9(1937),, 224–248. 10(1938), 110–163.Google Scholar
14. Rahman, Q.I., Interpolation of entire functions, Amer. J., Math. 87(1965), 1029–1076.Google Scholar
15. Rahman, Q.I. and Schmeisser, G., IP inequalities for entire functions of exponential type, Trans. Amer. Math., Soc. 320(1990), 91–103.Google Scholar
16. Ronkin, L.I., Introduction to the theory of entire functions of several variables, Amer. Math. Soc. Transl. of Math. Monographs, Rhode, Island 44(1974).Google Scholar
17. Titchmarsh, E.C., The theory of functions, 2d ed.Oxford University Press, 1939.Google Scholar