Regularity of dual Gabor windows. (English) Zbl 1470.42058

Christensen, Ole; Kim, Hong Oh; Kim, Rae Young

Abstr. Appl. Anal. 2013, Article ID 747268, 8 p. (2013).

Summary: We present a construction of dual windows associated with Gabor frames with compactly supported windows. The size of the support of the dual windows is comparable to that of the given window. Under certain conditions, we prove that there exist dual windows with higher regularity than the canonical dual window. On the other hand, there are cases where no differentiable dual window exists, even in the overcomplete case. As a special case of our results, we show that there exists a common smooth dual window for an interesting class of Gabor frames. In particular, for any value of \(K \in \mathbb{N}\), there is a smooth function \(h\) which simultaneously is a dual window for all B-spline generated Gabor frames \(\{E_{m b} T_n B_N(x / 2) \}_{m, n \in \mathbb{N}}\) for B-splines \(B_N\) of order \(N = 1, \ldots, 2 K + 1\) with a fixed and sufficiently small value of \(b\).

MSC:

42C15	General harmonic expansions, frames
42A65	Completeness of sets of functions in one variable harmonic analysis

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