Non-Lebesgue multiresolution analyses
Classical notions of wavelets and multiresolution analyses deal with the Hilbert space $L^2(\mathbb R)$ and the standard translation and dilation operators. Key in the study of these subjects is the low-pass filter, which is a periodic function $h \in L^2([0,1))$ that satisfies the classical quadrature mirror filter equation $|h(x)|^2+|h(x+1/2)|^2 =2.$ This equation is satisfied almost everywhere with respect to Lebesgue measure on the torus. Generalized multiresolution analyses and wavelets exist in abstract Hilbert spaces with more general translation and dilation operators. Moreover, the concept of the low-pass filter has been generalized in various ways. It may be a matrix-valued function, it may not satisfy any obvious analog of a filter equation, and it may be an element of a non-Lebesgue $L^2$ space. In this article we discuss the last of these generalizations, i.e., filters that are elements of non-Lebesgue $L^2$ spaces. We give examples of such filters, and we derive ageneralization of the filter equation.