Univariate tight wavelet frames of minimal support

Abstract

This work characterizes (dyadic homogeneous) wavelet frames for $$L^2({{\mathbb {R}}})$$ by means of spectral techniques. These techniques use decomposability properties of the frame operator in spectral representations associated with the dilation operator. The approach is closely related to usual Fourier domain fiberization techniques, dual Gramian analysis, and extension principles. Spectral formulas are used to determine all the tight wavelet frames for $$L^2({{\mathbb {R}}})$$ with a fixed finite number of generators of minimal support. The method associates wavelet frames of this type with certain inner operator-valued functions in Hardy spaces. The cases with one and two generators are completely solved.