Note on B-splines, wavelet scaling functions, and Gabor frames

Let g be a continuous, compactly supported function on R such that the integer translates of g constitute a partition of unity. We show that the Gabor system (g, a, b), with window g and time-shift and frequency-shift parameters a, b > 0 has no lower frame bound larger than 0 if b = 2, 3, . . . and a > 0. In particular, (g, a, b) is not a Gabor frame if g is a continuous, compactly supported wavelet scaling function and if b = 2, 3, . . . and a > 0. We exemplify our result for the case that g = B1, the triangle function supported by [−1, 1], by showing pictures of the canonical dual corresponding to (g, a, b) when ab = 1/4 and b crosses the lines N = 2, 3, . . . . Keywords—Gabor frame, B-splines, wavelet scaling function, partition of unity, Ron-Shen condition.