On the characterization of scaling functions on non-Archimedean fields. (English) Zbl 1473.42042

Summary: In real life application all signals are not obtained from uniform shifts; so there is a natural question regarding analysis and decompositions of these types of signals by a stable mathematical tool. This gap was filled by J.-P. Gabardo and M. Z. Nashed [J. Funct. Anal. 158, No. 1, 209–241 (1998; Zbl 0910.42018)] by establishing a constructive algorithm based on the theory of spectral pairs for constructing non-uniform wavelet basis in \(L^2(\mathbb R)\). In this setting, the associated translation set \(\Lambda =\left\{ 0,r/N\right\}+2\,\mathbb Z\) is no longer a discrete subgroup of \(\mathbb R\) but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. In this paper, we characterize the scaling function for non-uniform multiresolution analysis on local fields of positive characteristic (LFPC). Some properties of wavelet scaling function associated with non-uniform multiresolution analysis (NUMRA) on LFPC are also established.


42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42C15 General harmonic expansions, frames
94A12 Signal theory (characterization, reconstruction, filtering, etc.)


Zbl 0910.42018
Full Text: DOI MNR


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