Shift Invariant Spaces and Shift Generated Dual Frames for Local Fields

Authors

  • A. Ahmadi Department of Mathematics, Hormozgan University, Bandar Abbas
  • A. Askari Hemmat Department of Mathematics,Shahid Bahonar University of Kerman; Department of Mathematics, Kerman Graduate University of technology, Kerman; International Center for science High Technology and Environment Science, Kerman

DOI:

https://doi.org/10.26713/cma.v3i3.205

Keywords:

Dual frames, Locally compact Abelian group, Shift invariant space, SG-dual frame

Abstract

Let $G$ be a locally compact Abelian group with a compact open subgroup $H$ and $X$ be a shift invariant subspace of $L^2(G)$ which forms a frame for a closed subspace of $L^2(G)$, then the dual frame of $X$ which is a shift invariant space, is called shift generated dual frame. In the present paper, we first define shift generated dual frame of type I and type II for a locally compact Abelian group with a compact open subgroup. Next, we present a characterization of shift generated dual frame in terms of fibers $H^\perp$.

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References

A.Ahmadi, A. Askari Hemmat and R. Raisi Tousi, Shift invariant spaces for local fields, Int. J. Wavelets Multiresolut. Inf. Process. 9(3) (2011), 417–426.

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

A. Askari Hemmat and J.P. Gabardo, The uniqueness of shift-generated duals for frames in shift-invariant subspaces, J. Fourier Anal. App. 13(5) (2007), 589–606.

J.J. Benedetto and R.L. Benedetto, A wavelet theory for local fields and related groups, J. Geom. Anal. 14(3) (2004), 423–456.

R.L. Benedetto, Examples of wavelets for local fields, in Wavelets, Frames, and Operator Theory, (College Park, MD, 2003), Am. Math. Soc. 27–47, Providence, RI, (2004).

M. Bownik, The structure of shift invariant subspaces of $L^2(R^n)$, J. Functional Anal. 177(2000), 282–309.

O. Christensen, An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, (2003).

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

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

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

H. Helson, Lectures on Invariant Subspaces, Academic Press, New York - London, (1964).

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

R.A. Kamyabi Gol and R. Raisi Tousi, A range function approach to shift invariant spaces on locally compact abelian group, Int. J.Wavelets, Multiresolut., Inf. Process (2010), 49–59.

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

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CITATION

How to Cite

Ahmadi, A., & Hemmat, A. A. (2012). Shift Invariant Spaces and Shift Generated Dual Frames for Local Fields. Communications in Mathematics and Applications, 3(3), 205–214. https://doi.org/10.26713/cma.v3i3.205

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Section

Research Article