Gabor Frames in L2(R)

In this paper, we introduce the notion of Walsh shift-invariant space and present a unified approach to the study of shift-invariant systems to be frames in L2(ℝ+). We obtain a necessary condition and three sufficient conditions under which the Walsh shift-invariant systems constitute frames for L2(ℝ+). Furthermore, we discuss applications of our main results to obtain some known conclusions about the Gabor frames and wavelet frames on positive half line.

In this article, fractional Gabor frames in L2(ℝ) are defined and studied. Also, a finite linear combination of fractional Gabor frames for L2(ℝ) is discussed. It is proved that under certain conditions, a finite linear combination of fractional Gabor frames is a fractional Gabor frame for L2(ℝ). Finally, the stability of fractional Gabor frames is discussed, and some results for the stability of fractional Gabor frames are proved.

It is shown that the Gabor frame operator for frames with lattice such thatab=1/N for some integerN≥1 is continuous on the modulation spaces ([9], [20]) whenever the basic functiong is a Schwartz function and (g, a, b) generates a frame forL2 (R). From this a sufficient condition on the lattice is given which guarantees that one has a Banach frame for the modulation spaces of Feichtinger.