Abstract:

The theme is to smooth characteristic functions of Parseval frame wavelet sets 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^{2} 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^{2}(R^{d}) increases and converges as d increases. On the other hand, we also construct sequences of smoothed Parseval frame wavelet sets which do have frame bounds which converge to 1. While these functions are explicitly defined, they cannot be the result of convolution.