PERTURBATION OF NONHARMONIC FOURIER SERIES AND NONUNIFORM SAMPLING THEOREM

Title & Authors
PERTURBATION OF NONHARMONIC FOURIER SERIES AND NONUNIFORM SAMPLING THEOREM
Park, Hee-Chul; Shin, Chang-Eon;

Abstract
For an entire function f whose Fourier transform has a compact support confined to $\small{[-{\pi},\;{\pi}]}$ and restriction to $\small{{\mathbb{R}}}$ belongs to $\small{L^2{\mathbb{R}}}$, we derive a nonuniform sampling theorem of Lagrange interpolation type with sampling points $\small{{\lambda}_n{\in}{\mathbb{R}},\;n{\in}{\mathbb{Z}}}$, under the condition that $\small{\frac{lim\;sup}{n{\rightarrow}{\infty}}|{\lambda}_n-n|}$<$\small{\frac {1}{4}}$.
Keywords
Riesz basis;frame;nonuniform sampling;nonharmonic Fourier series;
Language
English
Cited by
1.
Sampling and recovery of multidimensional bandlimited functions via frames, Journal of Mathematical Analysis and Applications, 2010, 367, 2, 374
2.
Multivariate polynomial interpolation and sampling in Paley–Wiener spaces, Journal of Approximation Theory, 2012, 164, 4, 460
3.
Perturbed sampling formulas and local reconstruction in shift invariant spaces, Journal of Mathematical Analysis and Applications, 2011, 377, 2, 841
References
1.
R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341-366

2.
I. M. Gel'fand and G. E. Shilov, Generalized functions. Vol. I: Properties and operations, Translated by Eugene Saletan Academic Press, New York-London, 1964

3.
J. R. Higgins, Sampling Theoy in Fourier and Signal Analysis, Oxford University Press Inc., New York, 1996

4.
J. R. Higgins, A sampling theorem for irregularly spaced sample points, IEEE Trans. Information Theory IT-22 (1976), no. 5, 621-622

5.
M. I. Kadec, The exact value of the Paley-Wiener constant, Dokl. Akad. Nauk SSSR 155 (1964), 1253-1254

6.
N. Levinson, Gap and Density Theorems, American Mathematical Society Colloquium Publications, v. 26. American Mathematical Society, New York, 1940

7.
R. E. A. C. Paley and N. Wiener, Fourier transforms in the complex domain, Reprint of the 1934 original. American Mathematical Society Colloquium Publications, 19. American Mathematical Society, Providence, RI, 1987

8.
R. M. Young, An introduction to nonharmonic Fourier series, Pure and Applied Mathematics, 93. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980