SHIFT GENERATED DUAL FRAMES FOR LOCALLY COMPACT ABELIAN GROUPS

Title & Authors
SHIFT GENERATED DUAL FRAMES FOR LOCALLY COMPACT ABELIAN GROUPS

Abstract
Let $\small{G}$ be a metrizable, $\small{{\sigma}}$-compact locally compact abelian group with a compact open subgroup. In this paper we define the Gramian and the dual Gramian operators for shift invariant subspaces of $\small{L^2(G)}$ and we use them to characterize shift generated dual frames for shift in- variant spaces, which forms a frame for a subspace of $\small{L^2(G)}$. We present necessary and sufficient conditions for which standard dual is a unique SG-dual frame of type I and type II.
Keywords
frames;Gramian operator;locally compact abelian group;shift invariant space;SG-dual frame;
Language
English
Cited by
References
1.
A. Ahmadi, A. A. Hemmat, and R. R. Tousi, Shift invariant spaces for local fields, Int. J. Wavelets Multiresolut. Inf. Process. 9 (2011), no. 3, 417-426.

2.
A. Ahmadi, A. A. Hemmat, and R. R. Tousi, A characterization of shift invariant spaces on LCA group G with a compact open subgroup, preprint.

3.
J. J. Benedetto and R. L. Benedetto, A wavelet theory for local elds and related groups, J. Geom. Anal. 14 (2004), no. 3, 423-456.

4.
R. L. Benedetto, Examples of wavelets for local elds, Wavelets, frames and operator theory, 27-47, Contemp. Math., 345, Amer. Math. Soc., Providence, RI, 2004.

5.
M. Bownik, The structure of shift invariant subspaces of $L^{2}(R^{n})$, J. Funct. Anal. 177 (2000), no. 2 282-309.

6.
O. Christensen, An Introduction to Frames and Riesz Bases, Birkhauser, Boston, 2003.

7.
I. Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal expansion, J. Math. Phys. 27 (1986), no. 5, 1271-1283.

8.
R. J. Duffin and A. C. Schaefer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341-366.

9.
G. B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, 1995.

10.
P. H. Frampton and Y. Okada, P-adic string N-point function, Phys. Rev. Lett. B 60 (1988), 484-486.

11.
H. Helson, Lectures on Invariant Subspaces, Academic Press, New York, London, 1964.

12.
A. A. Hemmat and J. P. Gabardo, The uniqueness of shift-generated duals for frames in shift-invariant subspaces, J. Fourier Anal. Appl. 13 (2007), no. 5, 589-606.

13.
E. Hewitt and K. A. Ross, Abstract Harmonic Analysis. Vol. I: Structure of Topological Groups. Integration Theory, Group Representations, Springer, Berlin, 1963.

14.
R. A. Kamyabi Gol and R. R. Tousi, The structure of shift invariant spaces on locally compact abelian group, J. Math. Anal. Appl. 340 (2008), 219-225.

15.
R. A. Kamyabi Gol and R. R. Tousi, A range function approach to shift invariant spaces on locally compact abelian group, Int. J. Wavelets Multiresolut. Inf. Process 8 (2010), no. 1, 49-59.

16.
N. J. Munch, Noise reduction in tight Weyl-Heisenberg frames, IEEE Trans. Inform. Theory 38 (1992), no. 2, part 2, 608-616.

17.
H. Reiter and J. D. Stegeman, Classical Harmonic Analysis and Locally Compact Groups, Clarendon Press. Oxford, 2000.

18.
W. Rudin, Real and Complex Analysis, McGraw-Hill Co., Singapore, 1987.