B-Gabor type frames in separable Hilbert spaces

Thomas Jineesh, N M Madhavan Namboothiri, P N Jayaprasad

Abstract


It is guaranteed that Gabor like structured frames do exist in any finite dimensional Hilbert space via an invertible map from l2(ZN) J Thomas and Nambudiri [2022]. Hence the question: whether it is possible to obtain structured class of frames in separable Hilbert spaces? is relevant. In this article we obtain a structured class of frames for separable Hilbert spaces which are generalizations of Gabor frames for L2(R) in their construction aspects. We call them as B-Gabor type frames and present a characterization of the frame operators associated with these frames when B is a unitary map. Some significant properties of the associated frame operators are discussed.


Keywords


Gabor frame; frame operator; translation; modulation

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References


P. Cazassa and G. Kutyniok. "Frames of subspaces, wavelets, frames and operator theory." American Mathematical Society, Contemporary Mathematics Publishers, 345:87–113, 2004.

O. Christensen. "An Introduction to Frames and Riesz Bases," 2nd edn. Birkhäuser, Boston, 2016.

O. Christensen and Y. Eldar. "Oblique dual frames and shift-invariant spaces." Applied and Computational Harmonic Analysis, 17:48–68, 2004.

D. Gabor. "Theory of communication." Journal of the Institution of Electrical Engineers, 93:429–457, 1946.

D. L. D. Han. "Frames, Bases and Group representations." Memoirs of the American Mathematical Society, USA, 2000.

R. Duffin and A. Schaeffler. "A class of non-harmonic Fourier series." Transactions of the American Mathematical Society, 72:341–366, 1952.

K. Gröchenig. "Foundations of Time Frequency Analysis." Birkhäuser, Boston, 2001.

A. G. I. Daubechies and Y. Meyer. "Painless nonorthogonal expansions." Journal of Mathematical Physics, 27:1271–1283, 1986.

J. Laurence. "Linear independence of Gabor systems in finite dimensional vector spaces." Journal of Fourier Analysis and Applications, 11:715–726, 2005.

N. N. J Thomas and T. Nambudiri. "A class of structured frames in finite dimensional Hilbert spaces." Journal of Korean Society of Mathematical Education. Series B: Pure and Applied Mathematics, 28:321–334, 2022.

S. Li and H. Ogawa. "Pseudoframes for subspaces with applications." Journal of Fourier Analysis and Applications, 10:409–431, 2004.

M. Janssen. "Gabor representation of generalized functions." Journal of Mathematical Analysis and Applications, 83:377–394, 1981.

T. Nambudiri and K. Parthasarathy. "Generalised Weyl-Heisenberg frame operators." Bulletin Des Sciences Mathematiques, 136:44–53, 2012.

T. Nambudiri and K. Parthasarathy. "Characterization of Weyl-Heisenberg frame operators." Bulletin Des Sciences Mathematiques, 137:322–324, 2013.

T. Nambudiri and K. Parthasarathy. "Wavelet frame operators and admissible frames." Asian-European Journal of Mathematics, 11:1–13, 2021.




DOI: http://dx.doi.org/10.23755/rm.v48i0.1236

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Ratio Mathematica - Journal of Mathematics, Statistics, and Applications. ISSN 1592-7415; e-ISSN 2282-8214.