From Fourier Analysis to Wavelets
This text introduces the basic concepts of function spaces and operators, both from the continuous and discrete viewpoints. Fourier and Window Fourier Transforms are introduced and used as a guide to arrive at the concept of Wavelet transform.
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Jonas Gomes Luiz Velho
From Fourier Analysis to Wavelets Instituto de Matemática Pura e Aplicada
IMPA Monographs Volume 3
This series, jointly established by IMPA and Springer, publishes advanced monographs giving authoritative accounts of current research in any field of mathematics, with emphasis on those fields that are closer to the areas currently supported at IMPA. The series gives well-written presentations of the “state-of-the-art” in fields of mathematical research and pointers to future directions of research.
Series Editors Emanuel Carneiro, Instituto de Matemática Pura e Aplicada Severino Collier, Universidade Federal do Rio de Janeiro Claudio Landim, Instituto de Matemática Pura e Aplicada Paulo Sad, Instituto de Matemática Pura e Aplicada
More information about this series at http://www.springer.com/series/13501
Jonas Gomes • Luiz Velho
From Fourier Analysis to Wavelets
123
Luiz Velho Instituto de Matemática Pura e Aplicada Rio de Janeiro, Brazil
Jonas Gomes BOZANO Investimentos Offices Shopping Leblon Rio de Janeiro, Brazil
IMPA Monographs ISBN 978-3-319-22074-1 DOI 10.1007/978-3-319-22075-8
ISBN 978-3-319-22075-8 (eBook)
Library of Congress Control Number: 2015945959 Mathematics Subject Classification (2010): 42-XX, 42C40, 65-XX, 65T60 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www. springer.com)
To my family, the true force that propelled my carrier. — Jonas Gomes
To my beloved daughter Alice Velho Chekroun and my grandson Levi Yitzchak Chekroun — Luiz Velho.
Preface
From a digital viewpoint, most real-world applications can be reduced to the problem of function representation and reconstruction. These two problems are closely related to synthesis and analysis of functions. The Fourier transform is the classical
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