I will present some results from the newly developed theory of wavelets; based on the joint work with the following authors:
P.M. Luthy, H.Šikić, F.Soria, G.L.Weiss, E.N.Wilson.One-DimensionalDyadic Wavelets.Mem. Amer. Math. Soc. 280 (2022), no 1378, ix+152 pp.
About two and a half decades ago and based on the influential book by Fernandez and Weiss, an approach was developed to study wavelets from the point of view of their connections with Fourier analysis. The idea was to study wavelets and other reproducing function systems via the four basic equations that characterized various properties of wavelet systems, like frame and basis properties, completeness, orthogonality, etc. Despite hundreds of research papers and the impressive development of the theory of such systems, some questions remain open even in the basic case of dyadic wavelets on the real line. Among the most thorough treatments that we provide in this lengthy paper is the issue of the understanding of the low-pass filters associated with the MRA structure. In this talk, I will focus on some of these results. As it turned out, a more general and abstract approach to the problem, using the study of dyadic orbits and the newly introduced Tauberian function, reveals several interesting properties and opens an interesting context for some older results