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Examples of wavelets for local fields. (English) Zbl 1062.42027

Heil, Christopher (ed.) et al., Wavelets, frames and operator theory. Papers from the Focused Research Group Workshop, University of Maryland, College Park, MD, USA, January 15–21, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3380-4/pbk). Contemporary Mathematics 345, 27-47 (2004).
Summary: It is well known that the Haar and Shannon wavelets in \(L^2(\mathbb{R})\) are at opposite extremes, in the sense that the Haar wavelet is localized in time but not in frequency, whereas the Shannon wavelet is localized in freqency but not in time. We present a rich setting where the Haar and Shannon wavelets coincide and are localized both in time and in frequency.
More generally, if \(\mathbb{R}\) is replaced by a group \(G\) with certain properties, J. Benedetto and the author have proposed a theory of wavelets on \(G\), including the construction of wavelet sets [“A wavelet theory for local fields and related groups”, submitted]. Examples of such groups \(G\) include the \(p\)-adic rational group \(G =\mathbb{Q}_p\), which is simply the completion of \(\mathbb{Q}\) with respect to a certain natural metric topology, and the Cantor dyadic group \(\mathbb{F}_2((t))\) of formal Laurent series with coefficients 0 or 1.
In this expository paper, we consider some specific examples of the wavelet theory on such groups \(G\). In particular, we show that Shannon wavelets on \(G\) are the same as Haar wavelets on \(G\). We also give several examples of specific groups (such as \(\mathbb{Q}_p\) and \(\mathbb{F}_p((t))\), for any prime number \(p\)) and of various wavelets on those groups. All of our wavelets are localized in frequency; the Haar/Shannon wavelets are localized both in time and in frequency.
For the entire collection see [Zbl 1052.42002].

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
11S85 Other nonanalytic theory
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