Riesz and Tight Wavelet Frame Sets in Locally Compact Abelian Groups

In this paper, we attempt to obtain sufficient conditions for the existence of tight wavelet frame sets in locally compact abelian groups. The condition is generated by modulating a collection of characteristic functions that correspond to a generalized shift-invariant system via the Fourier transform. We present two approaches (for stationary and non-stationary wavelets) to construct the scaling function for L-2 (G) and, using the scaling function, we construct an orthonormal wavelet basis for L-2 (G). We propose an open problem related to the extension principle for Riesz wavelets in locally compact abelian groups.