On the \(p\)-adic spectral analysis and multiwavelet on \(L^2({\mathbb Q}_p^n)\). (English) Zbl 1033.42030

The basic operator in the analysis on complex-valued functions over non-Archimedean local fields is the fractional differential operator \(D^\alpha\) introduced by Vladimirov. This operator considered on \(L_2(\mathbb Q_p)\) has a pure point spectrum, with eigenvalues of infinite multiplicity. An explicit construction of an eigenbasis was first given by Vladimirov. Due to the infinite multiplicity, it is possible to construct eigenbases with different properties, and a new simpler construction was proposed recently by S. V. Kozyrev [Izv. Math. 66, No. 2, 367–376 (2002); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 66, No. 2, 149–158 (2002; Zbl 1016.42025)] and applied by him to the 2-adic interpretation of wavelets.
The paper under review is devoted to a multidimensional analog of Kozyrev’s results. The technique is a straightforward generalization of the one used in the one-dimensional case. Note that many calculations by the authors, which are longer than those in the one-dimensional case, could be avoided by taking into account that the operator considered by the authors is equivalent to a similar one-dimensional operator over an unramified extension of \(\mathbb Q_p\).


42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
47G30 Pseudodifferential operators


Zbl 1016.42025
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