\(p\)-adic wavelets (Q2825726)

This paper is a short survey on wavelet theory for functions of \(p\)-adic argument. As usual, let \(\mathbb{Q}_p\) be the field of \(p\)-adic numbers. The \(p\)-adic Haar basis in \(L^2(\mathbb{Q}_p)\) was constructed by Kozyrev in 2002. A number of \(p\)-adic wavelet bases and frames consisting of test functions (i.e. compactly supported band-limited functions) appeared later. A unique construction of a non-compactly supported orthogonal wavelet basis for \(L^2(\mathbb{Q}_p)\) was given in [\textit{S. Evdokimov}, J. Math. Anal. Appl. 443, No. 2, 1260--1266 (2016; Zbl 1344.42029)]. It is now known that any orthogonal \(p\)-adic wavelet basis consisting of test functions is a modification of the \(p\)-adic Haar basis. Moreover, it is proved that any orthogonal \(p\)-adic wavelet basis consisting of band-limited functions is a modification of an orthogonal basis generated by the \(p\)-adic Haar MRA (see [\textit{S. Evdokimov} and the author, J. Math. Anal. Appl. 424, No. 2, 952--965 (2015; Zbl 1304.42087)]). However, it is not known today whether there are \(p\)-adic wavelet bases or frames consisting of functions such that not all of them are band-limited. The question of why Haar bases in various structures are the same was discussed in [\textit{I. Novikov} and the author, Math. Notes 91, No. 6, 895--898 (2012; Zbl 1286.42053)]. While it is not known whether there is an orthogonal \(p\)-adic wavelet basis which is not a modification of an orthogonal basis generated by the \(p\)-adic Haar MRA, a variety of orthogonal wavelet bases generated by MRAs essentially different from the Haar MRA are constructed on other structures such as Vilenkin groups and local fields of positive characteristic (see, e.g., [\textit{Yu. A. Farkov} and \textit{E. A. Rodionov}, p-Adic Numbers Ultrametric Anal. Appl. 3, No. 3, 181--195 (2011; Zbl 1254.42044)]).