On the vaguelet and Riesz properties of L2L2-unbounded transformations of orthogonal wavelet bases

In this work, we prove that certain L2L2-unbounded transformations of orthogonal wavelet bases generate vaguelets. The L2L2-unbounded functions involved in the transformations are assumed to be quasi-homogeneous at high frequencies. We provide natural examples of functions which are not quasi-homogeneous and for which the resulting transformations are not vaguelets. We also address the related question of whether the considered family of functions is a Riesz basis in L2(R)L2(R). The Riesz property could be deduced directly from the results available in the literature or, as we outline, by using the vaguelet property in the context of this work. The considered families of functions arise in wavelet-based decompositions of stochastic processes with uncorrelated coefficients.