Introduction to Fourier analysis and wavelets

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Introduction to Fourier analysis and wavelets / Mark A. Pinsky

資料種別:: 図書
出版情報:: Australia : Brooks/Cole, c2002
形態:: xviii, 376 p. ; 25 cm
シリーズ名:: Brooks/Cole series in advanced mathematics <BA44745464>
著者名:: Pinsky, Mark A., 1940- <DA00247309>
ISBN:: 9780534376604 [0534376606]
書誌ID:: BA55295133

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注記:

Includes bibliographical references and index
"Thomson learning"

主題:

Fourier analysis; Wavelets (Mathematics)

言語:

英語

目次情報:

Fourier Series on the Circle / 1：

Motivation and Heuristics / 1.1：

Motivation from Physics / 1.1.1：

The Vibrating String / 1.1.1.1：

Heat Flow in Solids / 1.1.1.2：

Absolutely Convergent Trigonometric Series / 1.1.2：

Examples of Factorial and Bessel Functions / 1.1.3：

Poisson Kernel Example / 1.1.4：

Proof of Laplace's Method / 1.1.5：

Nonabsolutely Convergent Trigonometric Series / 1.1.6：

Formulation of Fourier Series / 1.2：

Fourier Coefficients and Their Basic Properties / 1.2.1：

Fourier Series of Finite Measures / 1.2.2：

Rates of Decay of Fourier Coefficients / 1.2.3：

Piecewise Smooth Functions / 1.2.3.1：

Fourier Characterization of Analytic Functions / 1.2.3.2：

Sine Integral / 1.2.4：

Other Proofs That Si([infinity]) = 1 / 1.2.4.1：

Pointwise Convergence Criteria / 1.2.5：

Integration of Fourier Series / 1.2.6：

Convergence of Fourier Series of Measures / 1.2.6.1：

Riemann Localization Principle / 1.2.7：

Gibbs-Wilbraham Phenomenon / 1.2.8：

The General Case / 1.2.8.1：

Fourier Series in L[superscript 2] / 1.3：

Mean Square Approximation--Parseval's Theorem / 1.3.1：

Application to the Isoperimetric Inequality / 1.3.2：

Rates of Convergence in L[superscript 2] / 1.3.3：

Application to Absolutely-Convergent Fourier Series / 1.3.3.1：

Norm Convergence and Summability / 1.4：

Approximate Identities / 1.4.1：

Almost-Everywhere Convergence of the Abel Means / 1.4.1.1：

Summability Matrices / 1.4.2：

Fejer Means of a Fourier Series / 1.4.3：

Wiener's Closure Theorem on the Circle / 1.4.3.1：

Equidistribution Modulo One / 1.4.4：

Hardy's Tauberian Theorem / 1.4.5：

Improved Trigonometric Approximation / 1.5：

Rates of Convergence in C (T) / 1.5.1：

Approximation with Fejer Means / 1.5.2：

Jackson's Theorem / 1.5.3：

Higher-Order Approximation / 1.5.4：

Converse Theorems of Bernstein / 1.5.5：

Divergence of Fourier Series / 1.6：

The Example of du Bois-Reymond / 1.6.1：

Analysis via Lebesgue Constants / 1.6.2：

Divergence in the Space L[superscript 1] / 1.6.3：

Appendix: Complements on Laplace's Method / 1.7：

First Variation on the Theme-Gaussian Approximation / 1.7.0.1：

Second Variation on the Theme-Improved Error Estimate / 1.7.0.2：

Application to Bessel Functions / 1.7.1：

The Local Limit Theorem of DeMoivre-Laplace / 1.7.2：

Appendix: Proof of the Uniform Boundedness Theorem / 1.8：

Appendix: Higher-Order Bessel functions / 1.9：

Appendix: Cantor's Uniqueness Theorem / 1.10：

Fourier Transforms on the Line And Space / 2：

Basic Properties of the Fourier Transform / 2.1：

Riemann-Lebesgue Lemma / 2.2.1：

Approximate Identities and Gaussian Summability / 2.2.2：

Improved Approximate Identities for Pointwise Convergence / 2.2.2.1：

Application to the Fourier Transform / 2.2.2.2：

The n-Dimensional Poisson Kernel / 2.2.2.3：

Fourier Transforms of Tempered Distributions / 2.2.3：

Characterization of the Gaussian Density / 2.2.4：

Wiener's Density Theorem / 2.2.5：

Fourier Inversion in One Dimension / 2.3：

Dirichlet Kernel and Symmetric Partial Sums / 2.3.1：

Example of the Indicator Function / 2.3.2：

Dini Convergence Theorem / 2.3.3：

Extension to Fourier's Single Integral / 2.3.4.1：

Smoothing Operations in R[superscript 1]-Averaging and Summability / 2.3.5：

Averaging and Weak Convergence / 2.3.6：

Cesaro Summability / 2.3.7：

Approximation Properties of the Fejer Kernel / 2.3.7.1：

Bernstein's Inequality / 2.3.8：

One-Sided Fourier Integral Representation / 2.3.9：

Fourier Cosine Transform / 2.3.9.1：

Fourier Sine Transform / 2.3.9.2：

Generalized h-Transform / 2.3.9.3：

L[superscript 2] Theory in R[superscript n] / 2.4：

Plancherel's Theorem / 2.4.1：

Bernstein's Theorem for Fourier Transforms / 2.4.2：

The Uncertainty Principle / 2.4.3：

Uncertainty Principle on the Circle / 2.4.3.1：

Spectral Analysis of the Fourier Transform / 2.4.4：

Hermite Polynomials / 2.4.4.1：

Eigenfunction of the Fourier Transform / 2.4.4.2：

Orthogonality Properties / 2.4.4.3：

Completeness / 2.4.4.4：

Spherical Fourier Inversion in R[superscript n] / 2.5：

Bochner's Approach / 2.5.1：

Piecewise Smooth Viewpoint / 2.5.2：

Relations with the Wave Equation / 2.5.3：

The Method of Brandolini and Colzani / 2.5.3.1：

Bochner-Riesz Summability / 2.5.4：

A General Theorem on Almost-Everywhere Summability / 2.5.4.1：

Bessel Functions / 2.6：

Fourier Transforms of Radial Functions / 2.6.1：

L[superscript 2]-Restriction Theorems for the Fourier Transform / 2.6.2：

An Improved Result / 2.6.2.1：

Limitations on the Range of p / 2.6.2.2：

The Method of Stationary Phase / 2.7：

Statement of the Result / 2.7.1：

Proof of the Method of Stationary Phase / 2.7.2：

Abel's Lemma / 2.7.4：

Fourier Analysis in L[superscript p] Spaces / 3：

The M. Riesz-Thorin Interpolation Theorem / 3.1：

Generalized Young's Inequality / 3.2.0.1：

The Hausdorff-Young Inequality / 3.2.0.2：

Stein's Complex Interpolation Theorem / 3.2.1：

The Conjugate Function or Discrete Hilbert Transform / 3.3：

L[superscript p] Theory of the Conjugate Function / 3.3.1：

L[superscript 1] Theory of the Conjugate Function / 3.3.2：

Identification as a Singular Integral / 3.3.2.1：

The Hilbert Transform on R / 3.4：

L[superscript 2] Theory of the Hilbert Transform / 3.4.1：

L[superscript p] Theory of the Hilbert Transform, 1 [ p [ [infinity] / 3.4.2：

Applications to Convergence of Fourier Integrals / 3.4.2.1：

L[superscript 1] Theory of the Hilbert Transform and Extensions / 3.4.3：

Kolmogorov's Inequality for the Hilbert Transform / 3.4.3.1：

Application to Singular Integrals with Odd Kernels / 3.4.4：

Hardy-Littlewood Maximal Function / 3.5：

Application to the Lebesgue Differentiation Theorem / 3.5.1：

Application to Radial Convolution Operators / 3.5.2：

Maximal Inequalities for Spherical Averages / 3.5.3：

The Marcinkiewicz Interpolation Theorem / 3.6：

Calderon-Zygmund Decomposition / 3.7：

A Class of Singular Integrals / 3.8：

Properties of Harmonic Functions / 3.9：

General Properties / 3.9.1：

Representation Theorems in the Disk / 3.9.2：

Representation Theorems in the Upper Half-Plane / 3.9.3：

Herglotz/Bochner Theorems and Positive Definite Functions / 3.9.4：

Poisson Summation Formula And Multiple Fourier Series / 4：

The Poisson Summation Formula in R[superscript 1] / 4.1：

Periodization of a Function / 4.2.1：

Statement and Proof / 4.2.2：

Shannon Sampling / 4.2.3：

Multiple Fourier Series / 4.3：

Basic L[superscript 1] Theory / 4.3.1：

Pointwise Convergence for Smooth Functions / 4.3.1.1：

Representation of Spherical Partial Sums / 4.3.1.2：

Basic L[superscript 2] Theory / 4.3.2：

Restriction Theorems for Fourier Coefficients / 4.3.3：

Poisson Summation Formula in R[superscript d] / 4.4：

Simultaneous Nonlocalization / 4.4.1：

Application to Lattice Points / 4.5：

Kendall's Mean Square Error / 4.5.1：

Landau's Asymptotic Formula / 4.5.2：

Application to Multiple Fourier Series / 4.5.3：

Three-Dimensional Case / 4.5.3.1：

Higher-Dimensional Case / 4.5.3.2：

Schrodinger Equation and Gauss Sums / 4.6：

Distributions on the Circle / 4.6.1：

The Schrodinger Equation on the Circle / 4.6.2：

Recurrence of Random Walk / 4.7：

Applications to Probability Theory / 5：

Basic Definitions / 5.1：

The Central Limit Theorem / 5.2.1：

Restatement in Terms of Independent Random Variables / 5.2.1.1：

Extension to Gap Series / 5.3：

Extension to Abel Sums / 5.3.1：

Weak Convergence of Measures / 5.4：

An Improved Continuity Theorem / 5.4.1：

Another Proof of Bochner's Theorem / 5.4.1.1：

Convolution Semigroups / 5.5：

The Berry-Esseen Theorem / 5.6：

Extension to Different Distributions / 5.6.1：

The Law of the Iterated Logarithm / 5.7：

Introduction to Wavelets / 6：

Heuristic Treatment of the Wavelet Transform / 6.1：

Wavelet Transform / 6.2：

Wavelet Characterization of Smoothness / 6.2.0.1：

Haar Wavelet Expansion / 6.3：

Haar Functions and Haar Series / 6.3.1：

Haar Sums and Dyadic Projections / 6.3.2：

Completeness of the Haar Functions / 6.3.3：

Haar Series in C[subscript 0] and L[subscript p] Spaces / 6.3.3.1：

Pointwise Convergence of Haar Series / 6.3.3.2：

Construction of Standard Brownian Motion / 6.3.4：

Haar Function Representation of Brownian Motion / 6.3.5：

Proof of Continuity / 6.3.6：

Levy's Modulus of Continuity / 6.3.7：

Multiresolution Analysis / 6.4：

Orthonormal Systems and Riesz Systems / 6.4.1：

Scaling Equations and Structure Constants / 6.4.2：

From Scaling Function to MRA / 6.4.3：

Additional Remarks / 6.4.3.1：

Meyer Wavelets / 6.4.4：

From Scaling Function to Orthonormal Wavelet / 6.4.5：

Direct Proof that V[subscript 1] [minus sign in circle] V[subscript 0] Is Spanned by {[Psi](t - k)}[subscript k[set membership]Z] / 6.4.5.1：

Null Integrability of Wavelets Without Scaling Functions / 6.4.5.2：

Wavelets with Compact Support / 6.5：

From Scaling Filter to Scaling Function / 6.5.1：

Explicit Construction of Compact Wavelets / 6.5.2：

Daubechies Recipe / 6.5.2.1：

Hernandez-Weiss Recipe / 6.5.2.2：

Smoothness of Wavelets / 6.5.3：

A Negative Result / 6.5.3.1：

Cohen's Extension of Theorem 6.5.1 / 6.5.4：

Convergence Properties of Wavelet Expansions / 6.6：

Wavelet Series in L[superscript p] Spaces / 6.6.1：

Large Scale Analysis / 6.6.1.1：

Almost-Everywhere Convergence / 6.6.1.2：

Convergence at a Preassigned Point / 6.6.1.3：

Jackson and Bernstein Approximation Theorems / 6.6.2：

Wavelets in Several Variables / 6.7：

Two Important Examples / 6.7.1：

Tensor Product of Wavelets / 6.7.1.1：

General Formulation of MRA and Wavelets in R[superscript d] / 6.7.2：

Notations for Subgroups and Cosets / 6.7.2.1：

Riesz Systems and Orthonormal Systems in R[superscript d] / 6.7.2.2：

Scaling Equation and Structure Constants / 6.7.2.3：

Existence of the Wavelet Set / 6.7.2.4：

Proof That the Wavelet Set Spans V[subscript 1] [minus sign in circle] V[subscript 0] / 6.7.2.5：

Cohen's Theorem in R[superscript d] / 6.7.2.6：

Examples of Wavelets in R[superscript d] / 6.7.3：

References

Notations

Index

Fourier Series on the Circle / 1：

Motivation and Heuristics / 1.1：

Motivation from Physics / 1.1.1：

The Vibrating String / 1.1.1.1：

Heat Flow in Solids / 1.1.1.2：

Absolutely Convergent Trigonometric Series / 1.1.2：

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