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図書

図書
Kenneth B. Howell
出版情報: Boca Raton, FL : Chapman & Hall/CRC, c2001  776 p. ; 26 cm
シリーズ名: Studies in advanced mathematics
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Preface
Sample Courses
Preliminaries / I:
The Starting Point / 1:
Fourier's Bold Conjecture / 1.1:
Mathematical Preliminaries and the Following Chapters / 1.2:
Basic Terminology, Notation, and Conventions / 2:
Numbers / 2.1:
Functions, Formulas, and Variables / 2.2:
Operators and Transforms / 2.3:
Basic Analysis I: Continuity and Smoothness / 3:
(Dis)Continuity / 3.1:
Differentiation / 3.2:
Basic Manipulations and Smoothness / 3.3:
Addenda / 3.4:
Basic Analysis II: Integration and Infinite Series / 4:
Integration / 4.1:
Infinite Series (Summations) / 4.2:
Symmetry and Periodicity / 5:
Even and Odd Functions / 5.1:
Periodic Functions / 5.2:
Sines and Cosines / 5.3:
Elementary Complex Analysis / 6:
Complex Numbers / 6.1:
Complex-Valued Functions / 6.2:
The Complex Exponential / 6.3:
Functions of a Complex Variable / 6.4:
Functions of Several Variables / 7:
Basic Extensions / 7.1:
Single Integrals of Functions with Two Variables / 7.2:
Double Integrals / 7.3:
Addendum / 7.4:
Fourier Series / II:
Heuristic Derivation of the Fourier Series Formulas / 8:
The Frequencies / 8.1:
The Coefficients / 8.2:
Summary / 8.3:
The Trigonometric Fourier Series / 9:
Defining the Trigonometric Fourier Series / 9.1:
Computing the Fourier Coefficients / 9.2:
Partial Sums and Graphing / 9.3:
Fourier Series over Finite Intervals (Sine and Cosine Series) / 10:
The Basic Fourier Series / 10.1:
The Fourier Sine Series / 10.2:
The Fourier Cosine Series / 10.3:
Using These Series / 10.4:
Inner Products, Norms, and Orthogonality / 11:
Inner Products / 11.1:
The Norm of a Function / 11.2:
Orthogonal Sets of Functions / 11.3:
Orthogonal Function Expansions / 11.4:
The Schwarz Inequality for Inner Products / 11.5:
Bessel's Inequality / 11.6:
The Complex Exponential Fourier Series / 12:
Derivation / 12.1:
Notation and Terminology / 12.2:
Computing the Coefficients / 12.3:
Partial Sums / 12.4:
Convergence and Fourier's Conjecture / 13:
Pointwise Convergence / 13.1:
Uniform and Nonuniform Approximations / 13.2:
Convergence in Norm / 13.3:
The Sine and Cosine Series / 13.4:
Convergence and Fourier's Conjecture: The Proofs / 14:
Basic Theorem on Pointwise Convergence / 14.1:
Convergence for a Particular Saw Function / 14.2:
Convergence for Arbitrary Saw Functions / 14.3:
Derivatives and Integrals of Fourier Series / 15:
Differentiation of Fourier Series / 15.1:
Differentiability and Convergence / 15.2:
Integrating Periodic Functions and Fourier Series / 15.3:
Sine and Cosine Series / 15.4:
Applications / 16:
The Heat Flow Problem / 16.1:
The Vibrating String Problem / 16.2:
Functions Defined by Infinite Series / 16.3:
Verifying the Heat Flow Problem Solution / 16.4:
Classical Fourier Transforms / III:
Heuristic Derivation of the Classical Fourier Transform / 17:
Riemann Sums over the Entire Real Line / 17.1:
The Derivation / 17.2:
Integrals on Infinite Intervals / 17.3:
Absolutely Integrable Functions / 18.1:
The Set of Absolutely Integrable Functions / 18.2:
Many Useful Facts / 18.3:
Functions with Two Variables / 18.4:
The Fourier Integral Transforms / 19:
Definitions, Notation, and Terminology / 19.1:
Near-Equivalence / 19.2:
Linearity / 19.3:
Invertibility / 19.4:
Other Integral Formulas (A Warning) / 19.5:
Some Properties of the Transformed Functions / 19.6:
Classical Fourier Transforms and Classically Transformable Functions / 20:
The First Extension / 20.1:
The Set of Classically Transformable Functions / 20.2:
The Complete Classical Fourier Transforms / 20.3:
What Is and Is Not Classically Transformable? / 20.4:
Duration, Bandwidth, and Two Important Sets of Classically Transformable Functions / 20.5:
More on Terminology, Notation, and Conventions / 20.6:
Some Elementary Identities: Translation, Scaling, and Conjugation / 21:
Translation / 21.1:
Scaling / 21.2:
Practical Transform Computing / 21.3:
Complex Conjugation and Related Symmetries / 21.4:
Differentiation and Fourier Transforms / 22:
The Differentiation Identities / 22.1:
Rigorous Derivation of the Differential Identities / 22.2:
Higher Order Differential Identities / 22.3:
Anti-Differentiation and Integral Identities / 22.4:
Gaussians and Other Very Rapidly Decreasing Functions / 23:
Basic Gaussians / 23.1:
General Gaussians / 23.2:
Gaussian-Like Functions / 23.3:
Complex Translation and Very Rapidly Decreasing Functions / 23.4:
Convolution and Transforms of Products / 24:
Derivation of the Convolution Formula / 24.1:
Basic Formulas and Properties of Convolution / 24.2:
Algebraic Properties / 24.3:
Computing Convolutions / 24.4:
Existence, Smoothness, and Derivatives of Convolutions / 24.5:
Convolution and Fourier Analysis / 24.6:
Correlation, Square-Integrable Functions, and the Fundamental Identity of Fourier Analysis / 25:
Correlation / 25.1:
Square-Integrable/Finite Energy Functions / 25.2:
The Fundamental Identity / 25.3:
Identity Sequences / 26:
An Elementary Identity Sequence / 26.1:
General Identity Sequences / 26.2:
Gaussian Identity Sequences / 26.3:
Verifying Identity Sequences / 26.4:
An Application (with Exercises) / 26.5:
Generalizing the Classical Theory: A Naive Approach / 27:
Delta Functions / 27.1:
Transforms of Periodic Functions / 27.2:
Arrays of Delta Functions / 27.3:
The Generalized Derivative / 27.4:
Fourier Analysis in the Analysis of Systems / 28:
Linear, Shift-Invariant Systems / 28.1:
Computing Outputs for LSI Systems / 28.2:
Gaussians as Test Functions, and Proofs of Some Important Theorems / 29:
Testing for Equality with Gaussians / 29.1:
The Fundamental Theorem on Invertibility / 29.2:
The Fourier Differential Identities / 29.3:
The Fundamental and Convolution Identities of Fourier Analysis / 29.4:
Generalized Functions and Fourier Transforms / IV:
A Starting Point for the Generalized Theory / 30:
Starting Points / 30.1:
Gaussian Test Functions / 31:
The Space of Gaussian Test Functions / 31.1:
On Using the Space of Gaussian Test Functions / 31.2:
Two Other Test Function Spaces and a Confession / 31.3:
More on Gaussian Test Functions / 31.4:
Norms and Operational Continuity / 31.5:
Generalized Functions / 32:
Functionals / 32.1:
Basic Algebra of Generalized Functions / 32.2:
Generalized Functions Based on Other Test Function Spaces / 32.4:
Some Consequences of Functional Continuity / 32.5:
The Details of Functional Continuity / 32.6:
Sequences and Series of Generalized Functions / 33:
Sequences and Limits / 33.1:
A Little More on Delta Functions / 33.2:
Basic Transforms of Generalized Fourier Analysis / 33.4:
Fourier Transforms / 34.1:
Generalized Scaling of the Variable / 34.2:
Generalized Translation/Shifting / 34.3:
Transforms of Limits and Series / 34.4:
Adjoint-Defined Transforms in General / 34.6:
Generalized Complex Conjugation / 34.7:
Generalized Products, Convolutions, and Definite Integrals / 35:
Multiplication and Convolution / 35.1:
Definite Integrals of Generalized Functions / 35.2:
Appendix: On Defining Generalized Products and Convolutions / 35.3:
Periodic Functions and Regular Arrays / 36:
Periodic Generalized Functions / 36.1:
Fourier Series for Periodic Generalized Functions / 36.2:
On Proving Theorem 36.5 / 36.3:
General Solutions to Simple Equations and the Pole Functions / 37:
Basics on Solving Simple Algebraic Equations / 37.1:
Homogeneous Equations with Polynomial Factors / 37.2:
Nonhomogeneous Equations with Polynomial Factors / 37.3:
The Pole Functions / 37.4:
Pole Functions in Transforms, Products, and Solutions / 37.5:
The Discrete Theory / V:
Periodic, Regular Arrays / 38:
The Index Period and Other Basic Notions / 38.1:
Fourier Series and Transforms of Periodic, Regular Arrays / 38.2:
Sampling and the Discrete Fourier Transform / 39:
Some General Conventions and Terminology / 39.1:
Sampling and the Discrete Approximation / 39.2:
The Discrete Approximation and Its Transforms / 39.3:
The Discrete Fourier Transforms / 39.4:
Discrete Transform Identities / 39.5:
Fast Fourier Transforms / 39.6:
Appendices
Fourier Transforms of Some Common Functions / Table A.1:
Identities for the Fourier Transforms / Table A.2:
References
Answers to Selected Exercises
Index
Preface
Sample Courses
Preliminaries / I:
2.

図書

図書
Albert Boggess, Francis J. Narcowich
出版情報: Upper Saddle River, NJ : Prentice Hall, c2001  xix, 283 p. ; 25 cm
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Preface
Acknowledgments
Inner Product Spaces / 0:
Motivation / 0.1:
Definition of Inner Product / 0.2:
The Spaces L[superscript 2] and l[superscript 2] / 0.3:
Definitions / 0.3.1:
Convergence in L[superscript 2] versus Uniform Convergence / 0.3.2:
Schwarz and Triangle Inequalities / 0.4:
Orthogonality / 0.5:
Definitions and Examples / 0.5.1:
Orthogonal Projections / 0.5.2:
Gram-Schmidt Orthogonalization / 0.5.3:
Linear Operators and Their Adjoints / 0.6:
Linear Operators / 0.6.1:
Adjoints / 0.6.2:
Least Squares and Linear Predictive Coding / 0.7:
Best Fit Line for Data / 0.7.1:
General Least Squares Algorithm / 0.7.2:
Linear Predictive Coding / 0.7.3:
Exercises / 0.8:
Fourier Series / 1:
Introduction / 1.1:
Historical Perspective / 1.1.1:
Signal Analysis / 1.1.2:
Partial Differential Equations / 1.1.3:
Computation of Fourier Series / 1.2:
On the Interval -[pi] [less than or equal] x [less than or equal] [pi] / 1.2.1:
Other Intervals / 1.2.2:
Cosine and Sine Expansions / 1.2.3:
Examples / 1.2.4:
The Complex Form of Fourier Series / 1.2.5:
Convergence Theorems for Fourier Series / 1.3:
The Riemann-Lebesgue Lemma / 1.3.1:
Convergence at a Point of Continuity / 1.3.2:
Convergence at a Point of Discontinuity / 1.3.3:
Uniform Convergence / 1.3.4:
Convergence in the Mean / 1.3.5:
The Fourier Transform / 1.4:
Informal Development of the Fourier Transform / 2.1:
The Fourier Inversion Theorem / 2.1.1:
Properties of the Fourier Transform / 2.1.2:
Basic Properties / 2.2.1:
Fourier Transform of a Convolution / 2.2.2:
Adjoint of the Fourier Transform / 2.2.3:
Plancherel Formula / 2.2.4:
Linear Filters / 2.3:
Time Invariant Filters / 2.3.1:
Causality and the Design of Filters / 2.3.2:
The Sampling Theorem / 2.4:
The Uncertainty Principle / 2.5:
Discrete Fourier Analysis / 2.6:
The Discrete Fourier Transform / 3.1:
Definition of Discrete Fourier Transform / 3.1.1:
Properties of the Discrete Fourier Transform / 3.1.2:
The Fast Fourier Transform / 3.1.3:
The FFT Approximation to the Fourier Transform / 3.1.4:
Application--Parameter Identification / 3.1.5:
Application--Discretizations of Ordinary Differential Equations / 3.1.6:
Discrete Signals / 3.2:
Time Invariant, Discrete Linear Filters / 3.2.1:
Z-Transform and Transfer Functions / 3.2.2:
Haar Wavelet Analysis / 3.3:
Why Wavelets? / 4.1:
Haar Wavelets / 4.2:
The Haar Scaling Function / 4.2.1:
Basic Properties of the Haar Scaling Function / 4.2.2:
The Haar Wavelet / 4.2.3:
Haar Decomposition and Reconstruction Algorithms / 4.3:
Decomposition / 4.3.1:
Reconstruction / 4.3.2:
Filters and Diagrams / 4.3.3:
Summary / 4.4:
Multiresolution Analysis / 4.5:
The Multiresolution Framework / 5.1:
Definition / 5.1.1:
The Scaling Relation / 5.1.2:
The Associated Wavelet and Wavelet Spaces / 5.1.3:
Decomposition and Reconstruction Formulas: A Tale of Two Bases / 5.1.4:
Implementing Decomposition and Reconstruction / 5.1.5:
The Decomposition Algorithm / 5.2.1:
The Reconstruction Algorithm / 5.2.2:
Processing a Signal / 5.2.3:
Fourier Transform Criteria / 5.3:
The Scaling Function / 5.3.1:
Orthogonality via the Fourier Transform / 5.3.2:
The Scaling Equation via the Fourier Transform / 5.3.3:
Iterative Procedure for Constructing the Scaling Function / 5.3.4:
The Daubechies Wavelets / 5.4:
Daubechies's Construction / 6.1:
Classification, Moments, and Smoothness / 6.2:
Computational Issues / 6.3:
The Scaling Function at Dyadic Points / 6.4:
Other Wavelet Topics / 6.5:
Computational Complexity / 7.1:
Wavelet Algorithm / 7.1.1:
Wavelet Packets / 7.1.2:
Wavelets in Higher Dimensions / 7.2:
Relating Decomposition and Reconstruction / 7.3:
Transfer Function Interpretation / 7.3.1:
Wavelet Transform / 7.4:
Definition of the Wavelet Transform / 7.4.1:
Inversion Formula for the Wavelet Transform / 7.4.2:
Technical Matters / Appendix A:
Proof of the Fourier Inversion Formula / A.1:
Rigorous Proof of Theorem 5.17 / A.2:
Proof of Theorem 5.10 / A.2.1:
Proof of the Convergence Part of Theorem 5.23 / A.2.2:
Matlab Routines / Appendix B:
General Compression Routine / B.1:
Use of MATLAB's FFT Routine for Filtering and Compression / B.2:
Sample Routines Using MATLAB's Wavelet Toolbox / B.3:
MATLAB Code for the Algorithms in Section 5.2 / B.4:
Bibliography
Index
Preface
Acknowledgments
Inner Product Spaces / 0:
3.

図書

図書
Mark A. Pinsky
出版情報: Australia : Brooks/Cole, c2002  xviii, 376 p. ; 25 cm
シリーズ名: Brooks/Cole series in advanced mathematics
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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:
4.

図書

図書
Eric Stade
出版情報: Hoboken, N.J. : John Wiley & Sons, c2005  xxiv, 488 p. ; 25 cm
シリーズ名: Pure and applied mathematics
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Preface
Introduction
Fourier Coefficients and Fourier Series / 1:
Fourier Series and Boundary Value Problems / 2:
L2 Spaces: Optimal Contexts for Fourier Series / 3:
Sturm-Liouville Problems / 4:
A Splat and a Spike / 5:
Fourier Transforms and Fourier Integrals / 6:
Special Topics and Applications / 7:
Local Frequency Analysis and Wavelets / 8:
Appendix
References
Index
Preface
Introduction
Fourier Coefficients and Fourier Series / 1:
5.

図書

図書
Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin
出版情報: Heidelberg : Springer, c2011  xv, 523 p. ; 25 cm
シリーズ名: Die Grundlehren der mathematischen Wissenschaften ; 343
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Basic Analysis / 1:
Basic Real Anslysis / 1.1:
Holder and Convolution Inequslities / 1.1.1:
The Atomic Decomposition / 1.1.2:
Proof of Refined Young Inequslityp8 / 1.1.3:
A Bilinear Interpolation Theorem / 1.1.4:
A Linear Interpolation Result / 1.1.5:
The Hardy-Littlewood Maximal Function / 1.1.6:
The Fourier Transform / 1.2:
Fourier Transforms of Functions and the Schwartz Space / 1.2.1:
Tempered Distributions and the Fourier Transform / 1.2.2:
A Few Calculations of Fourier Transforms / 1.2.3:
Homogeneous Sobolev Spaces / 1.3:
Definition and Basic Properties / 1.3.1:
Sobolev Embedding in Lebesgue Spaces / 1.3.2:
The Limit Case Hd/2 / 1.3.3:
The Embedding Theorem in Hölder Spaces / 1.3.4:
Nonhomogeneous Sobolev Spaces on Rd / 1.4:
Embedding / 1.4.1:
A Density Theorem / 1.4.3:
Hardy Inequality / 1.4.4:
References and Remarks / 1.5:
Littlewood-Paley Theory / 2:
Functions with Compactly Supported Fourier Transforms / 2.1:
Bernstein-Type Lemmas / 2.1.1:
The Smoothing Effect of Heat Flow / 2.1.2:
The Action of a Diffeomorphism / 2.1.3:
The Effects of Some Nonlinear Functions / 2.1.4:
Dyadic Partition of Unity / 2.2:
Homogeneous Besov Spaces / 2.3:
Characterizations of Homogeneous Besov Spaces / 2.4:
Besov Spaces, Lebesgue Spaces, and Refined Inequalities / 2.5:
Homogeneous Paradifferential Calculus / 2.6:
Homogeneous Bony Decomposition / 2.6.1:
Action of Smooth Functions / 2.6.2:
Time-Space Besov Spaces / 2.6.3:
Nonhomogeneous Besov Spaces / 2.7:
Nonhomogeneous Paradifferential Calculus / 2.8:
The Bony Decomposition / 2.8.1:
The Paralinearization Theorem / 2.8.2:
Besov Spaces and Compact Embeddings / 2.9:
Commutator Estimates / 2.10:
Around the Space B&infty;,&infty;1 / 2.11:
Transport and Transport-Diffusion Equations / 2.12:
Ordinary Differential Equations / 3.1:
The Cauchy-Lipschitz Theorem Revisited / 3.1.1:
Estimates for the Flow / 3.1.2:
A Blow-up Criterion for Ordinary Differential Equations / 3.1.3:
Transport Equations: The Lipschitz Case / 3.2:
A Priori Estimates in General Besov Spaces / 3.2.1:
Refined Estimates in Besov Spaces with Index 0 / 3.2.2:
Solving the Transport Equation in Besov Spaces / 3.2.3:
Application to a Shallow Water Equation / 3.2.4:
Losing Estimates for Transport Equations / 3.3:
Linear Loss of Regularity in Besov Spaces / 3.3.1:
The Exponential Loss / 3.3.2:
Limited Loss of Regularity / 3.3.3:
A Few Applications / 3.3.4:
Transport-Diffusion Equations / 3.4:
A Priori Estimates / 3.4.1:
Exponential Decay / 3.4.2:
Quasilinear Symmetric Systems / 3.5:
Definition and Examples / 4.1:
Linear Symmetric Systems / 4.2:
The Well-posedness of Linear Symmetric Systems / 4.2.1:
Finite Propagation Speed / 4.2.2:
Further Well-posedness Results for Linear Symmetric Systems / 4.2.3:
The Resolution of Quasilinear Symmetric Systems / 4.3:
Paralinearization and Energy Estimates / 4.3.1:
Convergence of the Scheme / 4.3.2:
Completion of the Proof of Existence / 4.3.3:
Uniqueness and Continuation Criterion / 4.3.4:
Data with Critical Regularity and Blow-up Criteria / 4.4:
Critical Besov Regularity / 4.4.1:
A Refined Blow-up Crndition / 4.4.2:
Continuity of the Flow Map / 4.5:
The Incompressible Navier-Stokes System / 4.6:
Basic Facts Concerning the Navier-Stokes System / 5.1:
Well-posedness in Sobolev Spaces / 5.2:
A General Result / 5.2.1:
The Behavior of the Hd/2-1 Norm Near 0 / 5.2.2:
Results Related to the Structure of the System / 5.3:
The Particular Case of Dimension Two / 5.3.1:
The Case of Dimension Three / 5.3.2:
An Elementary Lp Approach / 5.4:
The Endpoint Space for Picard's Scheme / 5.5:
The Use of the L1-smoothing Effect of the Heat Flow / 5.6:
The Cannone-Meyer-Planchon Theorem Revisited / 5.6.1:
The Flow of the Solutions of the Navier-Stokes System / 5.6.2:
Anisotropic Viscosity / 5.7:
The Case of L2 Data with One Vertical Derivative in L2 / 6.1:
A Global Existence Result in Anisotropic Besov Spaces / 6.2:
Anisotropic Localization in Fourier Space / 6.2.1:
The Functional Framework / 6.2.2:
Statement of the Main Result / 6.2.3:
Some Technical Lemmas / 6.2.4:
The Proof of Existence / 6.3:
The Proof of Uniqueness / 6.4:
Euler System for Perfect Incompressible Fluids / 6.5:
Local Well-posedness Results for Inviscid Fluids / 7.1:
The Biot-Savart Law / 7.1.1:
Estimates for the Pressure / 7.1.2:
Another Formulation of the Euler System / 7.1.3:
Local Existence of Smooth Solutions / 7.1.4:
Uniqueness / 7.1.5:
Continuation Criteria / 7.1.6:
Global Existence Results in Dimension Two / 7.2:
Smooth Solutions / 7.2.1:
The Borderline Case / 7.2.2:
The Yudovich Theorem / 7.2.3:
The Inviscid Limit / 7.3:
Regularity Results for the Navier-Stokes System / 7.3.1:
The Smooth Case / 7.3.2:
The Rough Case / 7.3.3:
Viscous Vortex Patches / 7.4:
Results Related to Striated Regularity / 7.4.1:
A Stationary Estimate for the Velocity Field / 7.4.2:
Uniform Estimates for Striated Regularity / 7.4.3:
A Global Convergence Result for Striated Regularity / 7.4.4:
Application to Smooth Vortex Patches / 7.4.5:
Strichartz Estimates and Applications to Semilinear Dispersive Equations / 7.5:
Examples of Dispersive Estimates / 8.1:
The Dispersive Estimate for the Free Transport Equation / 8.1.1:
The Dispersive Estimates for the Schrdillger Equation / 8.1.2:
Integral of Oscillating Functions / 8.1.3:
Dispersive Estimates for the Wave Equation / 8.1.4:
The L2 Boundedness of Some Fourier Integral Operators / 8.1.5:
Billnear Methods / 8.2:
The Duality Method and the TT* Argument / 8.2.1:
Strichartz Estimates: The Case q > 2 / 8.2.2:
Strichartz Estimates: The Endpoint Case q = 2 / 8.2.3:
Application to the Cubic Semilinear Schrödinger Equation / 8.2.4:
Strichartz Estimates for the Wave Equation / 8.3:
The Basic Strichartz Estimate / 8.3.1:
The Refined Strichartz Estimate / 8.3.2:
The Qulntic Wave Equation in R3 / 8.4:
The Cubic Wave Equation in R3 / 8.5:
Solutions in H1 / 8.5.1:
Local and Global Well-posedness for Rough Data / 8.5.2:
The Nonlinear Interpolation Method / 8.5.3:
Application to a Class of Semilinear Wave Equations / 8.6:
Smoothing Effect in Quasilinear Wave Equations / 8.7:
A Well-posedness Result Based on an Energy Method / 9.1:
The Main Statement and the Strategy of its Proof / 9.2:
Refined Paralinearization of the Wave Equation / 9.3:
Reduction to a Microlocal Strichartz Estimate / 9.4:
Microlocal Strichartz Estimates / 9.5:
A Rather General Statement / 9.5.1:
Geometrical Optics / 9.5.2:
The Solution of the Eikonal Equation / 9.5.3:
The Transport Equation / 9.5.4:
The Approximation Theorem / 9.5.5:
The Proof of Theorem 9.16 / 9.5.6:
The Compressible Navier-Stokes System / 9.6:
About the Model / 10.1:
General Overview / 10.1.1:
The Barotropic Navier-Stokes Equations / 10.1.2:
Local Theory for Data with Critical Regularity / 10.2:
Scaling Invariance and Statement of the Main Result / 10.2.1:
Existence of a Local Solution / 10.2.2:
A Continuation Criterion / 10.2.4:
Local Theory for Data Bounded Away from the Vacuum / 10.3:
A Priori Estimates for the Linearized Momentum Equation / 10.3.1:
Global Existence for Small Data / 10.3.2:
Statement of the Results / 10.4.1:
A Spectral Analysis of the Linearized Equation / 10.4.2:
A Prioli Estimates for the Linearized Equation / 10.4.3:
Proof of Global Existence / 10.4.4:
The Incompressible Limit / 10.5:
Main Results / 10.5.1:
The Case of Small Data with Critical Regularity / 10.5.2:
The Case of Large Data with More Regularity / 10.5.3:
References / 10.6:
List of Notations
Index
Basic Analysis / 1:
Basic Real Anslysis / 1.1:
Holder and Convolution Inequslities / 1.1.1:
6.

図書

図書
Audrey Terras
出版情報: Cambridge, U.K. : Cambridge University Press, 1999  x, 442 p. ; 23 cm
シリーズ名: London Mathematical Society student texts ; 43
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Introduction
Cast of characters
Congruences and the quotient ring of the integers mod n / Part I:
The discrete Fourier transform on the finite circle / 1.2:
Graphs of Z/nZ, adjacency operators, eigenvalues / 1.3:
Four questions about Cayley graphs / 1.4:
Finite Euclidean graphs and three questions about their spectra / 1.5:
Random walks on Cayley graphs / 1.6:
Applications in geometry and analysis / 1.7:
The quadratic reciprocity law / 1.8:
The fast Fourier transform / 1.9:
The DFT on finite Abelian groups - finite tori / 1.10:
Error-correcting codes / 1.11:
The Poisson sum formula on a finite Abelian group / 1.12:
Some applications in chemistry and physics / 1.13:
The uncertainty principle / 1.14:
Fourier transform and representations of finite groups / Part II:
Induced representations / 2.2:
The finite ax + b group / 2.3:
Heisenberg group / 2.4:
Finite symmetric spaces - finite upper half planes Hq / 2.5:
Special functions on Hq - K-Bessel and spherical / 2.6:
The general linear group GL(2, Fq) / 2.7:
Selberg's trace formula and isospectral non-isomorphic graphs / 2.8:
The trace formula on finite upper half planes / 2.9:
The trace formula for a tree and Ihara's zeta function / 2.10:
Introduction
Cast of characters
Congruences and the quotient ring of the integers mod n / Part I:
7.

図書

図書
M.W. Wong
出版情報: New York : Springer, c1998  viii, 158 p. ; 24 cm
シリーズ名: Universitext
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8.

図書

図書
George Bachman, Lawrence Narici, Edward Beckenstein
出版情報: New York : Springer, c2000  ix, 505 p. ; 25 cm
シリーズ名: Universitext
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9.

図書

図書
Dinakar Ramakrishnan, Robert J. Valenza
出版情報: New York : Springer, c1999  xxi, 350 p. ; 25 cm
シリーズ名: Graduate texts in mathematics ; 186
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10.

図書

図書
Javier Duoandikoetxea ; translated and revised by David Cruz-Uribe
出版情報: Providence, R.I. : American Mathematical Society, c2001  xviii, 222 p. ; 26 cm
シリーズ名: Graduate studies in mathematics ; v. 29
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Fourier series and integrals
The Hardy-Littlewood maximal function
The Hilbert transform Singular integrals (I)
Singular integrals (II)
$H^1$ and $BMO$ Weighted inequalities
Littlewood-Paley theory and multipliers
The $T1$ theorem
Bibliography
Index
Fourier series and integrals
The Hardy-Littlewood maximal function
The Hilbert transform Singular integrals (I)
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