On the splitting trick and wavelet frame packets

This paper presents a detailed analysis of the "splitting trick" which splits, for example, the half-shifts of a function into the shifts of two functions. When a Riesz basis of a shift-invariant subspace is split, the optimal bounds of the resulting Riesz basis are obtained. Most importantly, by the splitting trick we built wavelet frame packets as orthogonal wavelet packets constructed by Coifman and Meyer. Their algorithms for finding best basis for a function also apply to our setting. Key words. frame, wavelet, wavelet frame, Riesz basis, shift-invariant space.