Description An Introduction to Wavelets is the first volume in a new series, WAVELET ANALYSIS AND ITS APPLICATIONS. This is an introductory
treatise on wavelet analysis, with an emphasis on spline wavelets and time-frequency analysis. Among the basic topics covered in this
book are time-frequency localization, integral wavelet transforms, dyadic wavelets, frames, spline-wavelets, orthonormal wavelet bases,
and wavelet packets. In addition, the author presents a unified treatment of nonorthogonal, semiorthogonal, and orthogonal wavelets.
This monograph is self-contained, the only prerequisite being a basic knowledge of function theory and real analysis. It is suitable
as a textbook for a beginning course on wavelet analysis and is directed toward both mathematicians and engineers who wish to learn about
the subject. Specialists may use this volume as a valuable supplementary reading to the vast literature that has already emerged in this
field.
Audience
Academics and researchers, research and development engineers in industry, and graduate-level students.
Contents
An Overview: From Fourier Analysis to Wavelet Analysis. The Integral Wavelet Transform and Time-Frequency Analysis. Inversion
Formulas and Duals. Classification of Wavelets. Multiresolution Analysis, Splines, and Wavelets. Wavelet Decompositions and Reconstructions.
Fourier Analysis: Fourier and Inverse Fourier Transforms. Continuous-Time Convolution and the Delta Function. Fourier
Transform of Square-Integrable Functions. Fourier Series. Basic Convergence Theory and Poisson's Summation Formula.
Wavelet Transforms
and Time-Frequency Analysis: The Gabor Transform. Short-Time Fourier Transforms and the Uncertainty Principle. The Integral
Wavelet Transform. Dyadic Wavelets and Inversions. Frames. Wavelet Series.
Cardinal Spline Analysis: Cardinal Spline
Spaces. B-Splines and Their Basic Properties. The Two-Scale Relation and an Interpolatory Graphical Display Algorithm. B-Net Representations
and Computation of Cardinal Splines. Construction of Spline Approximation Formulas. Construction of Spline Interpolation Formulas.
Scaling
Functions and Wavelets: Multiresolution Analysis. Scaling Functions with Finite Two-Scale Relations. Direct-Sum Decompositions
of L2(R). Wavelets and Their Duals. Linear-Phase Filtering. Compactly Supported Wavelets. Cardinal Spline-Wavelets: Interpolaratory Spline-Wavelets.
Compactly Supported Spline-Wavelets. Computation of Cardinal Spline-Wavelets. Euler-Frobenius Polynomials. Error Analysis in Spline-Wavelet
Decomposition. Total Positivity, Complete Oscillation, Zero-Crossings. Orthogonal Wavelets and Wavelet Packets: Examples of Orthogonal
Wavelets. Identification of Orthogonal Two-Scale Symbols. Construction of Compactly Supported Orthogonal Wavelets. Orthogonal Wavelet
Packets. Orthogonal Decomposition of Wavelet Series. Notes. References. Subject Index. Appendix.
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