Wavelet analysis on local fields of positive characteristic
The book details wavelet analysis on topological groups, in particular local fields having positive characteristic, also known as modular representations. Topological groups are introduced in Chapter 1 and an existence and uniqueness proof is given for Haar measures on locally compact groups, illustrated by several examples. Fourier transforms in \(L^2\) spaces over local fields are introduced and convergence properties are analyzed. Chapter 2 presents multiresolution analysis of local fields of positive characteristic \(K\) and the construction of wavelets over \(K\). Examples of wavelets on \(p\)-series fields are given for \(p=2\) and \(3\). Affine frames generated by a finite family of functions in \(L^2\) are described in Chapter 3 and different properties, e.g. translation invariance, are characterized. Chapter 4 then provides a different characterization of functions involved in wavelet analysis, e.g. scaling functions, and wavelets, over local fields of positive characteristics. Concepts of multi-wavelets are likewise described including the construction of examples and some explicit formulas are given. Biorthogonal wavelets over a local field are introduced in Chapter 5 and it is shown that under certain conditions they generate a Riesz basis. Wavelet and wavelet frame packets, a generalization of wavelets, are then constructed in Chapter 6 using a so-called splitting trick. Again this construction is given in the context of local fields. Biorthogonal wavelet packets are introduced and it is shown that they yield a Riesz basis. Unconditional bases over a local field \(K\) in \(L^p\) spaces using Haar wavelets are provided in Chapter 7. The properties of principal shift-invariant spaces are analyzed in Chapter 8. The results are then used to characterize wavelets on local fields of positive characteristics. Notes at the end of each chapter make interesting links with existing literature and establish, whenever possible, some connections between the different approaches.
scientific article; zbMATH DE number 7466563