Smooth functions associated with wavelet sets on R, d ≥ 1, and frame bound gaps

The theme is to smooth characteristic functions of Parseval frame wavelet sets by convolution in order to obtain implementable, computationally viable, smooth wavelet frames. We introduce the following: a new method to improve frame bound estimation; a shrinking technique to construct frames; and a nascent theory concerning frame bound gaps. The phenomenon of a frame bound gap occurs when certain sequences of functions, converging in L to a Parseval frame wavelet, generate systems with frame bounds that are uniformly bounded away from 1. We prove that smoothing a Parseval frame wavelet set wavelet on the frequency domain by convolution with elements of an approximate identity produces a frame bound gap. Furthermore, the frame bound gap for such frame wavelets in L(R) increases and converges as d increases. ∗The authors gratefully acknowledge significant mathematical insights from Marcin Bownik, Wojciech Czaja, Bin Han, and David Larson. The first named author was supported in part by ONR Grant N00014-02-1-0398 and AFOSR MURI Grant AFOSR-FA9550-05-1-0443. The second author was supported in part by a Department of Education GAANN Fellowship and a University of Maryland Graduate School Ann G. Wylie Dissertation Fellowship.