Diagonalization of homogeneous linear operators in biorthogonal wavelet bases
Fredrik Ekstedt, Martin Lindberg · 1997
We show how it is possible to diagonalize a certain class of homogeneous linear operators in a biorthogonal wavelet basis. Given a linear operator and a biorthogonal wavelet basis we construct a new biorthogonal wavelet basis such that by analyzing a function in the new basis and multiplying the wavelet coefficients by a scale dependent factor we get the wavelet coefficients of the transformed function in the original wavelet basis. Differentiation and integration, the Riesz potential and the Hilbert transform belong to this class of operators. Finally we generalize the method to several dimensions including non-separable bases.