Construction of Wavelets Through Walsh Functions

This book focuses on the fusion of wavelets and Walsh analysis, which involves non-trigonometric function series (or WalshFourier series). The primary objective of the book is to systematically present the basic properties of non-trigonometric orthonormal systems such as the Haar system, HaarVilenkin system, Walsh system, wavelet system and frame system, as well as updated results on the book's main theme.
Based on lectures that the authors presented at several international conferences, the notions and concepts introduced in this interdisciplinary book can be applied to any situation where wavelets and their variants are used. Most of the applications of wavelet analysis and Walsh analysis can be tried for newly constructed wavelets. Given its breadth of coverage, the book offers a valuable resource for theoreticians and those applying mathematics in diverse areas. It is especially intended for graduate students of mathematics and engineering andresearchers interested in applied analysis.

Focuses on the fusion of wavelets and Walsh analysis, involving non-trigonometric function series Presents the basic properties of non-trigonometric orthonormal systems Discusses the most important results of Walsh and Haar analysis Identifies the Haar wavelet as the simplest example of a wavelet, and the Walsh function as the first example of a wavelet packet

Auteur

YURI A. FARKOV is a distinguished scientist and professor at the Russian Presidential Academy of National Economy and Public Administration (RANEPA), Moscow, Russia. Earlier, he worked as a professor and head of the Department of Mathematics at the Russian State Geological Prospecting University, Moscow, Russia, from 1997 to 2014; a professor and head of the Department of Higher Mathematics at Dubna International University, Dubna, Russia, from 1996 to 2000; an associate professor at the Department of Higher Mathematics and Mathematical Modelling, Moscow State Geological Prospecting Academy, from 1988 to 1997; and a lecturer at the Department of Higher Mathematics, Moscow State University of Mechanical Engineering "Environmental and Chemical Engineering Institute", Moscow, from 1981 to 1988. In 2013, Prof. Farkov received his DSc-Doctor Fiz.- Mat. Nauk from the People's Friendship University of Russia on "Optimal Methods of Approximation of Function by Generalized Polynomials and Wavelets". In 1981, he received his PhD degree in Mathematics from Moscow Region Pedagogical Institute, on topic "Investigations of Asymptotic and Approximation Properties of Faber-Erokhin Basis Functions". From 1977 to 1980, he did his postgraduate studies at the Moscow Electro-Technical Engineering Institute; and in 1975, he received his BSc degree in Mathematics from Uralsk Pedagogical Institute, Kazakhstan. Professor Farkov is on the editorial boards of the journals American Journal of Computational Mathematics, Communications in Mathematics and Applications and International Journal of Education. He is a member of the Moscow Mathematical Society and American Mathematical Society. His research interests include mathematical analysis, wavelet theory, dyadic analysis and approximation theory.
PAMMY MANCHANDA is a senior professor of mathematics at Guru Nanak Dev University, Amritsar, India. She has attended and delivered talks and chaired sessions at reputed academic conferences and workshops across the world, including ICIAM (1999-2015) and ICM since 2002. She was invited twice to the Industrial Mathematics Group of Professor Helmut Neunzert, Kaiserslautern University, Germany, and visited the International Centre for Theoretical Physics (a UNESCO institution) at Trieste, Italy, many times to carry out her research activities. She was the joint secretary of the Indian Society of Industrial and Applied Mathematics (ISIAM) from 1999 until 2016 and after that, she is the secretary of the society. She has been actively engaged in organizing international conferences by the society. She is the managing editor of the Indian Journal of Industrial and Applied Mathematics (by ISIAM) and a member of the editorial board of the Springer's book series, Industrial and Applied Mathematics. With a research interest in functional analysis and applications (financial and meteorological data) and wavelet theory, Prof.Manchanda has published 54 research papers in several international journals of repute, edited 3 proceedings of international conferences of ISIAM and co-authored 3 books.

ABUL HASAN SIDDIQI is a distinguished scientist and professor emeritus at the School of Basic Sciences and Research at Sharda University, Greater Noida, India. He was also a visiting consultant at the International Centre for Theoretical Physics (ICTP), Trieste, Italy; Sultan Qaboos University, Muscat, Oman; MIMOS, Kuala Lumpur, Malaysia; and professor at several reputed universities including Aligarh Muslim University (Aligarh, India) and King Fahd University of Petroleum & Minerals (Dhahran, Saudi Arabia). He has a long association with ICTP (a UNESCO institution) in several capacities: short-time visitor, long-duration visitor, regular associate, guests of the director and senior associate. He was awarded the German Academic Exchange Fellowship thrice to carry out mathematical research in Germany. He has published more than 100 research papers jointly with his research collaborators, 5 books and edited proceedings of 9 international conferences, as well as supervised 29 PhD scholars. He is the founder secretary and the elected president of the Indian Society of Industrial and Applied Mathematics (ISIAM), which celebrated its silver jubilee year in January 2016. He is the editor-in-chief of Indian Journal of Industrial and Applied Mathematics (published by ISIAM) and the Springer's book series, Industrial and Applied Mathematics.

Résumé
"The monograph is well written and the main concepts are clearly explained and presented. The contents are a comprehensive collection of results published during the last decades. The material is accessible to advanced undergraduate and graduate students with a good background in functional and Fourier analysis. For researchers, the volume is attractive for the broad and detailed presentation of the subject." (Peter R. Massopust, Mathematical Reviews, November, 2019)
"This book has an extensive bibliography of nearly 200 items and will be a valuable reference source. Since most chapters have exercises it could also be used as a textbook in an advanced graduate course. This work is noteworthy for its thoroughness, clarity, and historical perspective." (Richard A. Zalik, zbMath 1418.42002, 2019)

Contenu
Chapter 1. Introduction to Walsh Analysis and Wavelets.- Chapter 2. Walsh-Fourier Series.- Chapter 3. HaarFourier Analysis.- Chapter 4. Construction of Dyadic Wavelets through Walsh Functions.- Chapter 5. Orthogonal And Periodic Wavelets On Vilenkin Groups.- Chapter 6. Haar-Vilenkin Wavelet.- Chapter 7. Construction Biorthogonal Wavelets and Frames.- Chapter 8. Wavelets associated with Nonuniform Multiresolution analysis on positive half line.- Chapter 9. Orthogonal Vector Valued Wavelets on R+.- Appendices.