Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

Sampling expansions for functions having values in a Banach space

Author(s): DeGuang Han; Ahmed I. Zayed
Journal: Proc. Amer. Math. Soc. 133 (2005), 3597-3607.
MSC (2000): Primary 46B15, 46B45; Secondary 94A20, 42C40
Posted: June 8, 2005
MathSciNet review: 2163595
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: A sampling expansion for vector-valued functions having values in a Banach space, together with an inversion formula, is derived. The proof uses the concept of framing models of Banach spaces that generalizes the notion of frames in Hilbert spaces. Two examples illustrating the results are given, one involving functions having values in $L^{p}[-\pi, \pi], 1<p\leq 2$ , and the second involving functions having values in $L^{p}(\mathbb{R} )$ for $1 < p< \infty.$

References:

1.: M.H. Annaby and A.I. Zayed, On the use of Green's function in sampling theory, J. Integral Equations and Applications, 10 (1998), 117-139. MR 1646834 (99g:34061)
2.: P. Casazza, D. Han and D. Larson, Frames for Banach spaces, Contemp. Math., 247, Amer. Math. Soc., Providence, RI, 1999, 149-182. MR 1738089 (2000m:46015)
3.: O. Christensen and D. Stoeva, -frames in separable Banach spaces, Adv. Comput. Math., 18 (2003), 117-126. MR 1968115 (2004b:42060)
4.: O. Christensen, Atomic decomposition via projective group representations, Rocky Mountain J. Math., 26 (1996), 1289-1312. MR 1447588 (98h:43004)
5.: O. Christensen and C. Heil, Perturbations of Banach frames and atomic decompositions, Math. Nachr 185 (1997), 33-47. MR 1452474 (98m:42061)
6.: H. Feichtinger, Atomic characterizations of modulation spaces through Gabor-type representations, Rocky Mountain J. Math., 19 (1989), 113-125. MR 1016165 (90h:46051)
7.: H. Feichtinger and K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions I, J. Funct. Anal., 86 (1989), 307-340. MR 1021139 (91g:43011)
8.: H. Feichtinger and K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions II, Monatsh. Math., 108 (1989), 129-148. MR 1026614 (91g:43012)
9.: K. Gröchenig, Describing functions: Atomic decompositions versus frames, Monatshefte fur Math., 112(1991), 1-42. MR 1122103 (92m:42035)
10.: M. Frazier and B. Jawerth, Decomposition of Besov Spaces, Indiana Univ. Math. J., 34 (1985), 777-799. MR 0808825 (87h:46083)
11.: H.P. Kramer, A generalized sampling theorem, J. Math. Phys. 38 (1959), 68-72. MR 0103786 (21:2550)
12.: N. Levinson, Gap and Density Theorems, Amer. Math. Soc. Colloq. Publs. Ser., Vol. 26, 1940. MR 0003208 (2:180d)
13.: M.A. Naimark, Linear Differential Operators I, George Harrap, London, 1967.
14.: G. Weiss and E. Hernandez, A First Course On Wavelets, CRC Press, Boca Raton, FL 1996. MR 1408902 (97i:42015)
15.: A. Zayed, Sampling in a Hilbert space, Proc. Amer. Math. Soc., 124 (1996), 3767-3776. MR 1343731 (97b:41007)
16.: A.I. Zayed, A new role of Green's function in interpolation and sampling theory, J. Math. Anal. Appl. 175 (1993), 222-238. MR 1216757 (94d:41011)
17.: A.I. Zayed, Advances in Shannon's Sampling Theory, CRC Press, Boca Raton, 1993. MR 1270907 (95f:94008)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46B15, 46B45, 94A20, 42C40

Retrieve articles in all Journals with MSC (2000): 46B15, 46B45, 94A20, 42C40

Additional Information:

DeGuang Han
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
Email: dhan@pegasus.cc.ucf.edu

Ahmed I. Zayed
Affiliation: Department of Mathematical Sciences, DePaul University, Chicago, Illinois 60614
Email: azayed@math.depaul.edu

DOI: 10.1090/S0002-9939-05-08163-3
PII: S 0002-9939(05)08163-3
Keywords: Framing models, Banach spaces, atomic decomposition, interpolation, the Whittaker-Shannon-Kotel'nikov sampling theorem, wavelet basis
Received by editor(s): November 21, 2003
Received by editor(s) in revised form: July 23, 2004
Posted: June 8, 2005
Communicated by: David R. Larson
Copyright of article: Copyright 2005, American Mathematical Society

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|