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Wavelet Analysis on Local Fields of Positive Characteristic

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This book discusses the theory of wavelets on local fields of positive characteristic. The discussion starts with a thorough introduction to topological groups and local fields. It then provides a proof of the existence and uniqueness of Haar measures on locally compact groups. It later gives several examples of locally compact groups and describes their Haar measures. The book focuses on multiresolution analysis and wavelets on a local field of positive characteristic. It provides characterizations of various functions associated with wavelet analysis such as scaling functions, wavelets, MRA-wavelets and low-pass filters. Many other concepts which are discussed in details are biorthogonal wavelets, wavelet packets, affine and quasi-affine frames, MSF multiwavelets, multiwavelet sets, generalized scaling sets, scaling sets, unconditional basis properties of wavelets and shift invariant spaces. 

Author(s): Biswaranjan Behera, Qaiser Jahan
Series: Indian Statistical Institute Series
Publisher: Springer
Year: 2021

Language: English
Pages: 350
City: Singapore

Preface
Acknowledgements
Contents
About the Authors
1 Local Fields
1.1 Topological Groups
1.2 Haar Measure
1.3 The p-Adic and p-Series Fields
1.3.1 Haar Measure on mathbbQp
1.3.2 The p-Series Fields
1.4 Classification and Properties of Local Fields
1.4.1 Dual Groups of Some Classical Groups
1.5 Fourier Transforms of L1-Functions
1.6 Fourier Transforms of L2-Functions
1.7 Fourier Transforms of Lp-Functions, 1 1.8 Test Functions and Distributions on Local Fields
1.9 Fourier Series on the Ring of Integers
1.9.1 The Dyadic Group 2ω or the Walsh-Paley Group
1.9.2 Convergence of the Fourier Series
1.9.3 Cesàro Summability
1.9.4 Norm Convergence of the Cesàro Means
1.9.5 Decay of the Fourier Coefficients
1.9.6 Absolutely Convergent Fourier Series
1.10 Characters of Local Fields
1.10.1 Characters of Local Fields of Positive Characteristic
1.10.2 Characters of Local Fields of Characteristic Zero
1.11 Notes
References
2 Multiresolution Analysis on Local Fields
2.1 Wavelets on mathbbR
2.2 MRA on Local Fields of Positive Characteristic
2.2.1 Construction of Wavelets From a Multiresolution Analysis
2.3 Intersection Triviality and Union Density Conditions
2.4 Wavelets on p-Series Fields
2.4.1 The Case p=2
2.4.2 The Case p=3
2.5 Notes
References
3 Affine, Quasi-affine and Co-affine Frames
3.1 Affine and Quasi-affine Frames
3.2 Affine and Quasi-affine Duals
3.3 Translation Invariant Sesquilinear Operators
3.4 Co-affine Systems
3.4.1 Co-affine Systems and Multiresolution Analysis
3.5 Notes
References
4 Characterizations of Functions Associated with Wavelet Analysis
4.1 Characterization of Scaling Functions
4.2 Characterization of Wavelets
4.3 Characterization of MRA-Wavelets
4.4 Characterization of Low-Pass Filters
4.5 MSF-Wavelets, Wavelet Sets and Scaling Sets
4.6 An Explicit Formula for MRA-Wavelets
4.6.1 Examples
4.7 Notes
References
5 Biorthogonal Wavelets
5.1 Riesz Bases of Translates
5.2 Riesz Multiresolution Analysis
5.3 Biorthogonality of the Wavelets
5.4 Notes
References
6 Wavelet Packets and Frame Packets
6.1 The Splitting Lemma
6.2 Construction of Wavelet Packets
6.3 Wavelet Frame Packets
6.4 Splitting Lemma for Wavelet Frame Packets
6.5 Biorthogonal Wavelet Packets
6.6 Notes
References
7 Wavelets as Unconditional Bases
7.1 Pointwise Convergence of the Wavelet Expansions
7.2 Unconditionality of the Haar Wavelets on Lp(K)
7.3 Greedy Basis Property in Lp(K)
7.4 The Hardy Space H1(K) on a Local Field
7.5 Unconditionality of the Wavelet Systems
7.6 Notes
References
8 Shift-Invariant Spaces and Wavelets
8.1 Principal Shift-Invariant Spaces
8.1.1 Properties of the Periodized Function
8.1.2 Frames, Parseval Frames and Riesz Bases
8.1.3 Minimality and Non-redundancy
8.1.4 Principal Shift-Invariant Spaces in Terms of Fourier Transform
8.2 Shift-Invariant Subspaces
8.3 A Characterization of Wavelets
8.4 Notes
References
Index