Construction of Nonuniform Wavelet Frames on Non-Archimedean Fields
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Construction of Nonuniform Wavelet Frames on Non-Archimedean Fields Owais Ahmad1
· Neyaz Ahmad1
Received: 16 December 2019 / Accepted: 25 November 2020 / © The Author(s), under exclusive licence to Springer Nature B.V. part of Springer Nature 2020
Abstract A constructive algorithm based on the theory of spectral pairs for constructing nonuniform wavelet basis in L2 (R) was considered by Gabardo and Nashed (J Funct. Anal. 158:209-241, 1998). In this setting, the associated translation set = {0, r/N} + 2 Z is no longer a discrete subgroup of 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. The main objective of this paper is to develop oblique and unitary extension principles for the construction nonuniform wavelet frames over non-Archimedean Local fields of positive characteristic. An example and some potential applications are also presented. Keywords Nonuniform wavelet frame · Fourier transform · Non-Archimedean local field · Extension principles Mathematics Subject Classification (2010) 42C40 · 42C15 · 43A70 · 11S85
1 Introduction Duffin and Schaeffer [16] introduced the concept of frame in seperable Hilbert space while dealing with some deep problems in non-harmonic Fourier series. Frames are basis-like systems that span a vector space but allow for linear dependency, which can be used to reduce noise, find sparse representations, or obtain other desirable features unavailable with orthonormal bases. An important example about frame is
Owais Ahmad [email protected] Neyaz Ahmad [email protected] 1
Department of Mathematics, National Institute of Technology, Srinagar-190006, Jammu and Kashmir, India
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Math Phys Anal Geom
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wavelet frame, which is obtained by translating and dilating a finite family of functions. To mention only a few references on wavelet frames, the reader is referred to [10–15] and many references therein. Multiresolution analysis is an important mathematical tool since it provides a natural framework for understanding and constructing discrete wavelet systems. The concept of MRA has been extended in various ways in recent years. These concepts are generalized to L2 Rd , to lattices different from Zd , allowing the subspaces of MRA to be generated by Riesz basis instead of orthonormal basis, admitting a finite number of scaling functions, replacing the dilation factor 2 by an integer M 2 or by an expansive matrix A ∈ GLd (R) as long as A ⊂ AZd . All these concepts are developed on regular lattices, that is the translation set is always a group. Recently, Gabardo and Nashed [20] considered a generalization of Mallat’s [34] celebrated theory of MRA based on spectral pairs, in which the translation set acting on the scaling function associated with the MRA to generate the subspace V0 is no longer a group, but is the union of Z and a translate of Z. Based on one-dimensional spectral pairs, Gabardo and Yu [21] considered sets of nonuniform wavel
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