On WH Packets in $L^2(\mathbb{R})$

S. K. Kaushik, Ghanshyam Singh, . Virender

Abstract


WH packets with respect to a Gabor system (frame) have been introduced and it has been shown with the help of examples that WH packets with respect to a Gabor frame may not be a frame for $L^2(\mathbb{R})$. A sufficient condition under which WH packets with respect to a Gabor frame is a frame for $L^2(\mathbb{R})$ has been given. A necessary and sufficient condition has also been given in this direction. Further, finite sum of Gabor frames has been considered and sufficient conditions under which finite sum of Gabor frames is a frame for $L^2(\mathbb{R})$ have been given. Finally, stability of Gabor frames has been studied and sufficient conditions in this direction have been obtained.

Keywords


WH packets

Full Text:

PDF

References


P.G. Casazza, The art of frame theory,Taiwanese J. Math.4(2) (2000), 129–201.

O. Christensen,An Introduction to Frames and Riesz Bases, Birkhäuser, 2002.

O. Christensen, Frames perturbations,Proc. Amer. Math. Soc. 123 (1995), 1217–1220.

O. Chirstensen, A Paley-Wiener theorem for frames,Proc. Amer. Math. Soc. 123 (1995), 2199–2202.

I. Daubechies, A. Grossmann and Y. Meyer, Painless non-orthognal expansions, J. Math. Physics 27 (1986), 1271–1283.

R.J. Duffin and A.C. Schaeffer, A class of non-harmonic Fourier Series, Trans. Amer. Math. Soc. 72 (1952), 341–366.

S.J. Favier and R.A. Zalik, On stability of frames and Riesz bases, Appl. Comp. Harm. Anal. 2(1995), 160–173.

D. Gabor, Theory of communications, J. IEE, London 93(3) (1946), 429–457.

C. Heil and D.Walnut, Continuous and Discrete Wavelet transform, SIAM Review 31 (1989), 628–666.

R.E.A.C. Paley and N. Wiener, Fourier Transforms in the Complex Domain, AMS Colloquium Publications, 19, 1934.




DOI: http://dx.doi.org/10.26713%2Fcma.v3i3.216

Refbacks

  • There are currently no refbacks.


eISSN 0975-8607; pISSN 0976-5905