Mathematics Textbooks for Scienceand Engineering

Volume 2

For further volumes:http://www.springer.com/series/10785

Charles K. Chui • Qingtang Jiang

Applied Mathematics

Data Compression, Spectral Methods, FourierAnalysis, Wavelets, and Applications

Charles K. ChuiDepartment of StatisticsStanford UniversityStanford, CAUSA

Qingtang JiangDepartment of Mathematics

and Computer ScienceUniversity of MissouriSt. Louis, MOUSA

ISBN 978-94-6239-008-9 ISBN 978-94-6239-009-6 (eBook)DOI 10.2991/978-94-6239-009-6

Library of Congress Control Number: 2013939577Published by Atlantis Press, Paris, France www.atlantis-press.com

� Atlantis Press and the authors 2013This book, or any parts thereof, may not be reproduced for commercial purposes in any form or by anymeans, electronic or mechanical, including photocopying, recording or any information storage andretrieval system known or to be invented, without prior permission from the Publisher.

Printed on acid-free paper

Series Information

Textbooks in the series ‘Mathematics Textbooks for Science and Engineering’ willbe aimed at the broad mathematics, science and engineering undergraduate andgraduate levels, covering all areas of applied and applicable mathematics, inter-preted in the broadest sense.

Series Editor

Charles K. ChuiStanford University, Stanford, CA, USA

Atlantis Press8 square des Bouleaux75019 Paris, France

For more information on this series and our other book series, please visit ourwebsite www.atlantis-press.com

Editorial

Recent years have witnessed an extraordinarily rapid advance in the direction ofinformation technology within both the scientific and engineering disciplines. Inaddition, the current profound technological advances of data acquisition devicesand transmission systems contribute enormously to the continuing exponentialgrowth of data information that requires much better data processing tools. Tomeet such urgent demands, innovative mathematical theory, methods, and algo-rithms must be developed, with emphasis on such application areas as complexdata organization, contaminated noise removal, corrupted data repair, lost datarecovery, reduction of data volume, data dimensionality reduction, data com-pression, data understanding and visualization, as well as data security andencryption.

The revolution of the data information explosion as mentioned above demandsearly mathematical training with emphasis on data manipulation at the collegelevel and beyond. The Atlantis book series, ‘‘Mathematics Textbooks for Scienceand Engineering (MTSE)’’, is founded to meet the needs of such mathematicstextbooks that can be used for both classroom teaching and self-study. For thebenefit of students and readers from the interdisciplinary areas of mathematics,computer science, physical and biological sciences, and various engineering spe-cialties, contributing authors are requested to keep in mind that the writings for theMTSE book series should be elementary and relatively easy to read, with sufficientexamples and exercises. We welcome submission of such book manuscripts fromall who agree with us on this point of view.

This second volume is intended to be a comprehensive textbook in ‘‘Contem-porary Applied Mathematics’’, with emphasis in the following five areas: spectralmethods with applications to data analysis and dimensionality reduction; Fouriertheory and methods with applications to time-frequency analysis and solution ofpartial differential equations; Wavelet time-scale analysis and methods, with in-depth study of the lifting schemes, wavelet regularity theory, and convergence ofcascade algorithms; Computational algorithms, including fast Fourier transform,fast cosine transform, and lapped transform; and Information theory with appli-cations to image and video compression. This book is self-contained, with writing

vii

style friendly towards the teacher and reader. It is intended to be a textbook,suitable for teaching in a variety of courses, including: Applied Mathematics,Applied Linear Algebra, Applied Fourier Analysis, Wavelet Analysis, and Engi-neering Mathematics, both at the undergraduate and beginning graduate levels.

Charles K. ChuiMenlo Park, CA

viii Editorial

Preface

Mathematics was coined the ‘‘queen of science’’ by the ‘‘prince ofmathematicians,’’ Carl Friedrich Gauss, one of the greatest mathematicians of alltime. Indeed, the name of Gauss is associated with essentially all areas of math-ematics. It is therefore safe to assume that to Gauss there was no clear boundarybetween ‘‘pure mathematics’’ and ‘‘applied mathematics’’. To ensure financialindependence, Gauss decided on a stable career in astronomy, which is one of theoldest sciences and was perhaps the most popular one during the eighteenth andnineteenth centuries. In his study of celestial motion and orbits and a diversity ofdisciplines later in his career, including (in chronological order): geodesy, mag-netism, dioptrics, and actuarial science, Gauss has developed a vast volume ofmathematical methods and tools that are still instrumental to our current study ofapplied mathematics.

During the twentieth century, with the exciting development of quantum fieldtheory, with the prosperity of the aviation industry, and with the bullish activity infinancial market trading, and so forth, much attention was paid to the mathematicalresearch and development in the general discipline of partial differential equations(PDEs). Indeed, the non-relativistic modeling of quantum mechanics is describedby the Schrödinger equation; the fluid flow formulation, as an extension ofNewtonian physics by incorporating motion and stress, is modeled by the Navier-Stokes equation; and option stock trading with minimum risk can be modeled bythe Black-Scholes equation. All of these equations are PDEs. In general, PDEs areused to describe a wide variety of phenomena, including: heat diffusion, soundwave propagation, electromagnetic wave radiation, vibration, electrostatics, elec-trodynamics, fluid flow, and elasticity, just to name a few. For this reason, thetheoretical and numerical development of PDEs has been considered the core ofapplied mathematics, at least in the academic environment.

On the other hand, over the past two decades, we have witnessed a rapidlyincreasing volume of ‘‘information’’ contents to be processed and understood.With the recent advances of various high-tech fields and the popularity of socialnetworking, the trend of exponential growth of easily accessible information iscertainly going to continue well into the twenty-first century, and the bottleneckcreated by this information explosion will definitely require innovative solutions

ix

from the scientific and engineering communities, particularly those technologistswith better understanding of, and strong background in, applied mathematics.Today, ‘‘big data’’ is among the most pressing research directions. ‘‘Big data’’research and development initiatives have been created by practically all Federaldepartments and agencies of the United States. It is noted, in particular, that onMay 29, 2012, the Obama administration unveiled a ‘‘big data’’ initiative,announcing $200 million new R&D investments to help solve some of the nation’smost pressing challenges, by improving our ability to extract knowledge andinsights from large and complex collections of digital data. To launch the initia-tive, six Federal departments and agencies later announced more than $200 millionin new commitments that promised to greatly improve the tools and techniquesneeded to access, organize, and glean discoveries from huge volumes of digitaldata. ‘‘Mathematics of big data’’ is therefore expected to provide innovativetheory, methods, and algorithms to virtually every discipline, far beyond sciencesand engineering, for processing, transmitting, receiving, understanding, andvisualizing datasets, which could be very large or live in some high-dimensionalspaces.

Of course the basic mathematical tools, particularly PDE models and methods,are always among the core of the mathematical tool-box of applied mathematics.But other theory and methods have been integrated in this tool-box as well. One ofthe most essential ideas is the notion of ‘‘frequency’’ of the data information.A contemporary of Gauss, by the name of Joseph Fourier, instilled this importantconcept to our study of physical phenomena by his innovation of trigonometricseries representations, along with powerful mathematical theory and methods,which significantly expanded the core of the tool-box of applied mathematics. Thefrequency content of a given dataset facilitates the processing and understanding ofthe data information. Another important idea is the ‘‘multi-scale’’ structure ofdatasets. Less than three decades ago, with the birth of another exciting mathe-matical subject, called ‘‘wavelets’’, the dataset of information can be put in thewavelet domain for multi-scale processing as well. About half of this book isdevoted to the study of the theories, methods, algorithms, and computationalschemes of Fourier and wavelet analyses. In addition, other mathematical topics,which are essential to information processing but not commonly taught in a regularapplied mathematics course, are discussed in this book. These include informationcoding, data dimensionality reduction, and data compression.

The objective of this textbook is to introduce the basic theory and methods inthe tool-box of the core of applied mathematics, with a central scheme thataddresses information processing with emphasis on manipulation of digital imagedata. Linear algebra is presented as linear analysis, with emphasis on spectralrepresentation and principal component analysis (PCA), and with applications todata estimation and data dimensionality reduction. For data compression, thenotion of entropy is introduced to quantify coding efficiency as governed byShannon’s Noiseless Coding theorem. Discrete Fourier transform (DFT), followedby the discussion of an efficient computational algorithm, called fast Fouriertransform (FFT), as well as a real-valued version of the DFT, called discrete cosine

x Preface

transform (DCT), are studied, with application to extracting frequency content ofthe given discrete dataset that facilitates reduction of the entropy and thus sig-nificant improvement of the coding efficiency. While DFT is obtained from dis-cretization of the Fourier coefficient integral, four versions of discretization of theFourier cosine coefficients yield the DCT-I, DCT-II, DCT-III and DCT-IV that arecommonly used in applications. The integral version of DCT and Fourier coeffi-cient sequences is called the Fourier transform (FT). Analogous to the Fourierseries, the formulation of the inverse Fourier transform (IFT) is derived byapplying the Gaussian function as sliding time-window for simultaneous time-frequency localization, with optimality guaranteed by the Uncertainty Principle.Both the Fourier series and Fourier transform are applied to solving certain PDEs.

In addition, local time-frequency basis functions are introduced in this textbookby discretization of the frequency-modulated sliding time-window function at theinteger lattice points. Replacing the frequency modulation by modulation with thecosines avoids the Balian-Low stability restriction on the local time-frequency basisfunctions, with application to elimination of blocky artifacts caused by quantizationof tiled DCT in image compression. In Chap. 8, multi-scale data analysis isintroduced and compared with the Fourier frequency approach; the architecture ofmultiresolution approximation and analysis (MRA) is applied to the construction ofwavelets and formulation of the multi-scale wavelet decomposition and recon-struction algorithms; and the lifting scheme is also introduced to reduce the com-putational complexity of these algorithms as well as implementation of filter banks.The final two chapters of this book are devoted to an in-depth study of waveletanalysis, including construction of bi-orthogonal wavelets, their regularities, andconvergence of the cascade algorithms. The last three chapters alone, namelyChaps. 8–10, constitute a suitable textbook for a one-semester course in ‘‘WaveletAnalysis and Applications’’. For more details, a teaching guide is provided on pagesxvi–xix.

Most chapters of this Applied Mathematics textbook have been tested inclassroom teaching at both the undergraduate and graduate levels, and the finalrevision of the book manuscript was influenced by student feedback. The authorsare therefore thankful to have the opportunity to teach from the earlier drafts of thisbook in the university environment, particularly at the University of Missouri-St.Louis. In writing this textbook, the authors have benefited from assistance ofseveral individuals. In particular, they are most grateful to Margaret Chui for typingthe earlier versions of the first six chapters, to Tom Li for drawing many of thediagrams and improving the artworks, and to Maryke van der Walt for proofreadingthe entire book and suggesting several changes. In addition, the first author wouldlike to express his appreciation to the publisher, Atlantis Press, for their enthusiasmin publishing this book, and is grateful to Keith Jones and Zeger Karssen, inparticular, for their unconditional trust and lasting friendship over the years. Hewould also like to take this opportunity to acknowledge the generous support fromthe U.S. Army Research Office and the National Geospatial-Intelligence Agency of

Preface xi

his research in the broad mathematical subject of complex and high-dimensionaldata processing. The second author is indebted to his family for their sacrifice,patience, and understanding, during the long hours in preparing and writing thisbook.

Charles K. ChuiMenlo Park, California

Qingtang JiangSt. Louis Missouri

xii Preface

Contents

Teaching Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi

1 Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Sequence and Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . 141.3 Inner-Product Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.4 Bases of Sequence and Function Spaces. . . . . . . . . . . . . . . . . . 341.5 Metric Spaces and Completion . . . . . . . . . . . . . . . . . . . . . . . . 54

2 Linear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.1 Matrix Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642.2 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792.3 Eigenspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 892.4 Spectral Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3 Spectral Methods and Applications. . . . . . . . . . . . . . . . . . . . . . . . 1153.1 Singular Value Decomposition and Principal

Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1163.2 Matrix Norms and Low-Rank Matrix Approximation. . . . . . . . . 1323.3 Data Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1463.4 Data Dimensionality Reduction . . . . . . . . . . . . . . . . . . . . . . . . 157

4 Frequency-Domain Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1714.1 Discrete Fourier Transform. . . . . . . . . . . . . . . . . . . . . . . . . . . 1724.2 Discrete Cosine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 1794.3 Fast Fourier Transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1904.4 Fast Discrete Cosine Transform. . . . . . . . . . . . . . . . . . . . . . . . 199

5 Data Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2075.1 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

xiii

5.2 Binary Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2175.3 Lapped Transform and Compression Schemes. . . . . . . . . . . . . . 2315.4 Image and Video Compression . . . . . . . . . . . . . . . . . . . . . . . . 255

6 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2636.1 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2666.2 Fourier Series in Cosines and Sines . . . . . . . . . . . . . . . . . . . . . 2766.3 Kernel Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2886.4 Convergence of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . 2956.5 Method of Separation of Variables . . . . . . . . . . . . . . . . . . . . . 305

7 Fourier Time-Frequency Methods . . . . . . . . . . . . . . . . . . . . . . . . 3177.1 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3197.2 Inverse Fourier Transform and Sampling Theorem . . . . . . . . . . 3297.3 Isotropic Diffusion PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3397.4 Time-Frequency Localization . . . . . . . . . . . . . . . . . . . . . . . . . 3517.5 Time-Frequency Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3607.6 Appendix on Integration Theory . . . . . . . . . . . . . . . . . . . . . . . 373

8 Wavelet Transform and Filter Banks . . . . . . . . . . . . . . . . . . . . . . 3798.1 Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3818.2 Multiresolution Approximation and Analysis . . . . . . . . . . . . . . 3908.3 Discrete Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 4058.4 Perfect-Reconstruction Filter Banks . . . . . . . . . . . . . . . . . . . . . 419

9 Compactly Supported Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . 4339.1 Transition Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4369.2 Gramian Function G/ðxÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4459.3 Compactly Supported Orthogonal Wavelets . . . . . . . . . . . . . . . 4519.4 Compactly Supported Biorthogonal Wavelets . . . . . . . . . . . . . . 4659.5 Lifting Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

10 Wavelet Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49910.1 Existence of Refinable Functions in L2ðRÞ . . . . . . . . . . . . . . . . 50110.2 Stability and Orthogonality of Refinable Functions . . . . . . . . . . 51010.3 Cascade Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51710.4 Smoothness of Compactly Supported Wavelets . . . . . . . . . . . . . 535

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547

xiv Contents

Teaching Guide

This book is an elementary and yet comprehensive textbook that covers most ofthe standard topics in Applied Mathematics, with a unified theme of applications todata analysis, data manipulation, and data compression. It is a suitable textbook,not only for most undergraduate and graduate Applied Mathematics courses, butalso for a variety of Special Topics courses or seminars. The objective of this guideis to suggest several samples of such courses.

(1) A general ‘‘Applied Mathematics’’ course: Linear Spaces, Linear Analysis,Fourier Series, Fourier Transform, Partial Differential Equations, andApplications

(2) Applied Mathematics: with emphasis on computations and data compression(3) Applied Mathematics: with emphasis on data analysis and representation(4) Applied Mathematics: with emphasis on time-frequency analysis(5) Applied Mathematics: with emphasis on time-frequency and multi-scale

methods(6) Applied Linear Algebra(7) Applied Fourier Analysis(8) Applied and Computational Wavelet Analysis(9) Fourier and Wavelet Analyses

xv

1. Teaching Guide (for a general Applied Mathematics Course)

Chapter 1

Chapter 2

Chapter 3

Chapter 6

Chapter 7: 7.1, 7.2, 7.3

Chapters 4: 4.1, 4.2Supplementary materials on computations

2. Teaching Guide (Emphasis on computations and data compressions)

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 1: Background material, if needed

xvi Teaching Guide

3. Teaching Guide (Emphasis on data analysis and data representations)

Chapter 2

Chapter 3

Chapter 4

Chapter 6

Chapter 7: 7.1, 7.2, 7.3

Chapter 1: Background material, if needed

4. Teaching Guide (Emphasis on time-frequency analysis)

Chapter 6

Chapter 7: 7.1, 7.2, 7.4, 7.5

Chapter 8: 8.1 – 8.2

Chapter 4: 4.1 on background material, if needed

Teaching Guide xvii

5. Teaching Guide (Emphasis on time-frequency/time-scale methods)

Chapter 4: 4.1 – 4.2

Chapter 6: 6.1 – 6.2

Chapter 7: 7.1, 7.2, 7.4, 7.5

Chapter 8

Chapter 9: 9.5 only

6. Teaching Guide (Applied Linear Algebra)

Chapter 1: 1.2 – 1.5

Chapter 2

Chapter 3

Chapter 4

Chapter 5

7. Teaching Guide (Applied Fourier Analysis)

Chapter 6: 6.1 – 6.4

Chapter 4: 4.1 – 4.2

Chapter 5

Chapter 7: 7.1, 7.2, 7.4, 7.5

xviii Teaching Guide

8. Teaching Guide (Applied and Computational Wavelet Analysis)

Chapter 7: 7.1, 7.2, 7.4

Chapter 8

Chapter 9

Chapter 10

9. Teaching Guide (Fourier and Wavelet Analyses)

Chapter 6: 6.1 – 6.4

Chapter 7: 7.1, 7.2, 7.4, 7.5

Chapter 8

Chapter 9

Chapter 10

Teaching Guide xix

Figures

Fig. 1.1 Plots of c and �c in the complex planeFig. 1.2 Areas under y ¼ f ðxÞ and x ¼ f�1ðyÞ in the proof

of Young’s inequalityFig. 1.3 Orthogonal projection Pv in W

Fig. 3.1 Top Centered data (represented by �); Bottom original data(represented by �), dimension-reduced data (represented by �),and principal components v1; v2

Fig. 3.2 Top centered data (represented by �); bottom dimension-reduceddata (represented by �) with the first and second principalcomponents and original data (represented by �)

Fig. 4.1 Top real part of the DFT of S1; Bottom imaginary partof the DFT of S1

Fig. 4.2 Top real part of the DFT of S2; Bottom imaginarypart of the DFT of S2

Fig. 4.3 Top DCT of S1; Bottom DCT of S2

Fig. 4.4 FFT Signal flow chart for n ¼ 4Fig. 4.5 FFT signal flow chart for n ¼ 8Fig. 4.6 Lee’s fast DCT implementationFig. 4.7 Hou’s Fast DCT implementationFig. 4.8 Fast DCT implementation of Wang, Suehiro and HatoriFig. 5.1 Encoder : Q = quantization; E = entropy encodingFig. 5.2 Decoder : Q�1 = de-quantization; E�1 = de-codingFig. 5.3 Quantizers: low compression ratioFig. 5.4 Quantizers: high compression ratioFig. 5.5 Zig-zag orderingFig. 6.1 Coefficient plot of DnðxÞFig. 6.2 Dirichlet’s kernels DnðxÞ for n ¼ 4 (on left) and n ¼ 16 (on right)Fig. 6.3 Coefficient plot of rnðxÞFig. 6.4 Fejer’s kernels rnðxÞ for n ¼ 4 (on left) and n ¼ 16 (on right)Fig. 6.5 From top to bottom: f ðxÞ ¼ jxj and its 2p-extension,

S10f , S50f and S100f

xxi

Fig. 6.6 From top to bottom: f ðxÞ ¼ v½�p=2;p=2�ðxÞ;�p� x� pand its 2p-extension, S10f , S50f and S100f

Fig. 6.7 Gibbs phenomenonFig. 6.8 Insulated regionFig. 7.1 Diffusion with delta heat source (top) and arbitrary

heat source (bottom)Fig. 7.2 Admissible window function uðxÞ defined in (7.5.8)Fig. 7.3 Malvar wavelets w0;4 (top) and w0;16 (bottom).

The envelope of cosðk þ 12Þpx is the dotted graph of y ¼ �

ffiffiffi

2p

uðxÞFig. 8.1 Hat function / (on left) and its refinement (on right)Fig. 8.2 D4 scaling function / (on left) and wavelet w (on right)Fig. 8.3 Top: biorthogonal 5/3 scaling function / (on left)

and wavelet w (on right). Bottom: scaling function e/ (on left)

and wavelet ew (on right)Fig. 8.4 From top to bottom: original signal, details

after 1-, 2-, 3-level DWT, and the approximantFig. 8.5 From top to bottom: (modified) forward

and backward lifting algorithms of Haar filtersFig. 8.6 From top to bottom: (modified) forward and backward

lifting algorithms of 5/3-tap biorthogonal filtersFig. 8.7 Ideal lowpass filter (on left) and highpass filter (on right)Fig. 8.8 Decomposition and reconstruction algorithms with PR filter bankFig. 9.1 D6 scaling function / (on left) and wavelet w (on right)Fig. 9.2 D8 scaling function / (on left) and wavelet w (on right)Fig. 9.3 Top biorthogonal 9/7 scaling function / (on left) and wavelet w

(on right); Bottom scaling function e/ (on left) and wavelet ew(on right)

Fig. 10.1 /1 ¼ Qp/0;/2 ¼ Q2p/0;/3 ¼ Q3

p/0 obtained by cascadealgorithm with refinement mask of hat function and /0 ¼ v½0;1ÞðxÞ

Fig. 10.2 /1 ¼ Qp/0;/2 ¼ Q2p/0;/3 ¼ Q3

p/0 obtained by cascadealgorithm with refinement mask f1

2 ; 0;12g and /0 ¼ v½0;1ÞðxÞ

Fig. 10.3 /1 ¼ Qp/0;/4 ¼ Q4p/0 obtained by cascade algorithm with

refinement mask of / ¼ 13 v½0;3Þ and /0 being the hat function

Fig. 10.4 Piecewise linear polynomials /n ¼ Qnp/0; n ¼ 1; . . .; 4

approximating D4 scaling function / by cascade algorithm

xxii Figures