Wavelet Analysis and its Applications, Volume 1: An Introduction to Wavelets provides an introductory treatise on wavelet analysis with an emphasis on spline-wavelets and time-frequency analysis.
This book is divided into seven chapters. Chapter 1 presents a brief overview of the subject, including classification of wavelets, integral wavelet transform for time-frequency analysis, multi-resolution analysis highlighting the important properties of splines, and wavelet algorithms for decomposition and reconstruction of functions. The preliminary material on Fourier analysis and signal theory is covered in Chapters 2 and 3. Chapter 4 covers the introductory study of cardinal splines, while Chapter 5 describes a general approach to the analysis and construction of scaling functions and wavelets. Spline-wavelets are deliberated in Chapter 6. The last chapter is devoted to an investigation of orthogonal wavelets and wavelet packets.
This volume serves as a textbook for an introductory one-semester course on "wavelet analysis¿ for upper-division undergraduate or beginning graduate mathematics and engineering students.

Preface1. An Overview 1.1. Prom Fourier Analysis to Wavelet Analysis 1.2. the Integral Wavelet Transform and Time-Frequency Analysis 1.3. Inversion Formulas and Duals 1.4. Classification of Wavelets 1.5. Multi-Resolution Analysis, Splines, and Wavelets 1.6. Wavelet Decompositions and Reconstructions2. Fourier Analysis 2.1. Fourier and Inverse Fourier Transforms 2.2. Continuous-Time Convolution and the Delta Function 2.3. Fourier Transform of Square-Integrable Functions 2.4. Fourier Series 2.5. Basic Convergence Theory and Poisson'S Summation Formula3. Wavelet Transforms and Time-Frequency Analysis 3.1. the Gabor Transform 3.2. Short-Time Fourier Transforms and the Uncertainty Principle 3.3. the Integral Wavelet Transform 3.4. Dyadic Wavelets and Inversions 3.5. Frames 3.6. Wavelet Series4. Cardinal Spline Analysis 4.1. Cardinal Spline Spaces 4.2. ß-Splines and Their Basic Properties 4.3. The Two-Scale Relation and An Interpolatory Graphical Display Algorithm 4.4. B-Net Representations and Computation of Cardinal Splines 4.5. Construction of Spline Approximation Formulas 4.6. Construction of Spline Interpolation Formulas5. Scaling Functions and Wavelets 5.1. Multi-resolution Analysis 5.2. Scaling Functions with Finite Two-Scale Relations 5.3. Direct-Sum Decompositions of L2(R) 5.4. Wavelets and Their Duals 5.5. Linear-Phase Filtering 5.6. Compactly Supported Wavelets6. Cardinal Spline-Wavelets 6.1. Interpolatory Spline-Wavelets 6.2. Compactly Supported Spline-Wavelets 6.3. Computation of Cardinal Spline-Wavelets 6.4. Euler-Frobenius Polynomials 6.5. Error Analysis in Spline-Wavelet Decomposition 6.6. Total Positivity, Complete Oscillation, Zero-Crossings7. Orthogonal Wavelets and Wavelet Packets 7.1. Examples of Orthogonal Wavelets 7.2. Identification of Orthogonal Two-Scale Symbols 7.3. Construction of Compactly Supported Orthogonal Wavelets 7.4. Orthogonal Wavelet Packets 7.5. Orthogonal Decomposition of Wavelet SeriesNotesReferencesSubject IndexAppendix