|
Sampling in a Hilbert space
Author(s):
Ahmed
I.
Zayed
Journal:
Proc. Amer. Math. Soc.
124
(1996),
3767-3776.
MSC (1991):
Primary 41A05, 41A35;
Secondary 47A58
MathSciNet review:
1343731
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
An analog of the Whittaker-Shannon-Kotel'nikov sampling theorem is derived for functions with values in a separable Hilbert space. The proof uses the concept of frames and frame operators in a Hilbert space. One of the consequences of this theorem is that it allows us to derive sampling theorems associated with boundary-value problems and some homogeneous integral equations, which in turn gives us a generalization of another sampling theorem by Kramer.
References:
- [1]
- J. Benedetto, Frame decompositions, sampling and uncertainty principle inequalities, Wavelets: Mathematics and Applications, Editors, J. Benedetto and M. Frazier, CRC Press, Boca Raton, FL (1994), pp. 247-304. MR 94i:94005
- [2]
- -, Irregular sampling and frames, Wavelet-A Tutorial, Editor C. Chui, Academic Press (1991), pp. 1-63. MR 93c:42030
- [3]
- J. Benedetto and W. Heller, Irregular sampling and the theory of frames I, Note di Matematica, Vol. X, Suppl. 1 (1990), pp. 103-125. MR 94e:94006
- [4]
- F. Beutler, Error-free recovery of signals from irregularly spaced samples, SIAM Rev. 8 (1966), pp. 328-335. MR 34:5568
- [5]
- -, Sampling theorems and bases in a Hilbert space, Information and Control, 4 (1961), pp. 97-117. MR 26:4832
- [6]
- R. Boas, Entire Functions, Academic Press, New York (1954). MR 16:914f
- [7]
- C. Heil and D. Walnut, Continuous and discrete wavelet transforms, SIAM Rev., Vol. 31 (1989), pp. 628-666. MR 91c:42032
- [8]
- H. Kramer, A generalized sampling theorem, J. Math. Phys., 63 (1959/60), pp. 68-72. MR 21:2550
- [9]
- N. Levinson, Gap and Density Theorem, Amer. Math. Soc. Colloq. Publs., Ser., Vol. 26, Amer. Math. Soc., Providence, RI (1940). MR 2:180d
- [10]
- J. Mikusinski, The Bochner integral, Birkäuser, Basel (1978). MR 58:11296
- [11]
- R. Paley and N. Wiener, Fourier transforms in the complex domain, Amer. Math. Soc. Colloq. Publs., Ser., Vol. 19, Amer. Math. Soc., Providence, RI (1934). MR 31:5038
- [12]
- C. E. Shannon, A mathematical theory of communication, Bell System Tech. Report Journal, Vol. 27 (1948), pp. 379-423. MR 10:133e
- [13]
- K. Yao, Applications of reproducing kernel Hilbert space of band-limited signal models, Information and Control, Vol. 11 (1967), pp. 429-444.
- [14]
- A. I. Zayed, Advances in Shannon's sampling theory, CRC Press, Boca Raton, FL (1993). MR 95f:94008
- [15]
- -, A new role of Green's function in interpolation and sampling theory, J. Math. Anal. Appl., 175 (1993), pp. 272-238. MR 94d:41001
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical
Society
with
MSC (1991):
41A05, 41A35,
47A58
Retrieve articles in all Journals with
MSC (1991):
41A05, 41A35,
47A58
Additional Information:
Ahmed
I.
Zayed
Affiliation:
Department of Mathematics, University of Central Florida, Orlando, Florida 32816
Email:
fdzayed@ucf1vm.cc.ucf.edu
DOI:
10.1090/S0002-9939-96-03526-5
PII:
S 0002-9939(96)03526-5
Keywords:
Shannon's sampling theorem,
interpolation and approximation in a Hilbert space,
frames and frame operators
Received by editor(s):
May 30, 1994
Received by editor(s) in revised form:
June 5, 1995
Communicated by:
J. Marshall Ash
Copyright of article:
Copyright
1996,
American Mathematical Society
|
