Applied Analysis

Applied Analysis

by A.M. Krall

Paperback(Softcover reprint of the original 1st ed. 1986)

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Product Details

ISBN-13: 9789027723420
Publisher: Springer Netherlands
Publication date: 10/31/1986
Series: Mathematics and Its Applications , #31
Edition description: Softcover reprint of the original 1st ed. 1986
Pages: 561
Product dimensions: 6.10(w) x 9.25(h) x 0.05(d)

Table of Contents

I. Some Inequalities.- 1. Young’s Inequality.- 2. Hölder’s Inequality.- 3. Minkowski’s Inequality.- 4. A Relation between Different Norms.- II. Linear Spaces and Linear Operators.- 1. Linear Spaces.- 2. Linear Operators.- 3. Norms and Banach Spaces.- 4. Operator Convergence.- III. Existence and Uniqueness Theorems.- 1. The Contraction Mapping Theorem.- 2. Existence and Uniqueness of Solutions for Ordinary Differential Equations.- 3. First Order Linear Systems.- 4. n-th Order Differential Equations.- 5. Some Extensions.- IV. Linear Ordinary Differential Equations.- 1. First Order Linear Systems.- 2. Fundamental Matrices.- 3. Nonhomogeneous Systems.- 4. n-th Order Equations.- 5. Nonhomogeneous n-th Order Equations.- 6. Reduction of Order.- 7. Constant Coefficients.- V. Second Order Ordinary Differential Equations.- 1. A Brief Review.- 2. The Adjoint Operator.- 3. An Oscillation Theorem.- 4. The Regular Sturm-Liouville Problem.- 5. The Inverse Problem, Green’s Functions.- VI. The Stone-Weierstrass Theorem.- 1. Preliminary Remarks.- 2. Algebras and Subalgebras.- 3. The Stone-Weierstrass Theorem.- 4. Extensions and Special Cases.- VII. Hilbert Spaces.- 1. Hermitian Forms.- 2. Inner Product Spaces.- 3. Hilbert Spaces.- 4. Orthogonal Subspaces.- 5. Continuous Linear Functionals.- 6. Fourier Expansions.- 7. Isometric Hilbert Spaces.- VIII. Linear Operators on a Hilbert Space.- 1. Regular Operators on a Hilbert Space.- 2. Bilinear Forms, the Adjoint Operator.- 3. Self-Adjoint Operators.- 4. Projections.- 5. Some Spectral Theorems.- 6. Operator Convergence.- 7. The Spectral Resolution of a Self-Adjoint Operator.- 8. The Spectral Resolution of a Normal Operator.- 9. The Spectral Resolution of a Unitary Operator.- IX. Compact Operators on a Hilbert Space.- 1. Compact Operators.- 2. Some Special Examples.- 3. The Spectrum of a Compact Self-Adjoint Operator.- 4. The Spectral Resolution of a Compact, Self-Adjoint Operator.- 5. The Regular Sturm-Liouville Problem.- X. Special Functions.- 1. Orthogonal Polynomials.- 2. The Legendre Polynomials.- 3. The Laguerre Polynomials.- 4. The Hermite Polynomials.- 5. Bessel Functions.- XI. The Fourier Integral.- 1. The Lebesgue Integral.- 2. The Fourier Integral in L1(-?, ?).- 3. The Fourier Integral in L2(-?, ?).- XII. The Singular Sturm-Liouville Problem.- 1. Circles under Bilinear Transformations.- 2. Helly’s Convergence Theorems.- 3. Limit Points and Limit Circles.- 4. The Limit Point Case.- 5. The Limit Circle Case.- 6. Examples.- XIII. An Introduction to Partial Differential Equations.- 1. The Cauchy-Kowaleski Theorem.- 2. First Order Equations.- 3. Second Order Equations.- 4. Green’s Formula.- XIV. Distributions.- 1. Test Functions and Distributions.- 2. Limits of Distributions.- 3. Fourier Transforms of Distributions.- 4. Applications of Distributions to Ordinary Differential Equations.- 5. Applications of Distributions to Partial Differential Equations.- XV. Laplace’s Equation.- 1. Introduction, Well Posed Problems.- 2. Dirichlet, Neumann, and Mixed Boundary Value Problems.- 3. The Dirichlet Problem.- 4. The Dirichlet Problem on the Unit Circle.- 5. Other Examples.- XVI. The Heat Equation.- 1. Introduction, the Cauchy Problem.- 2. The Cauchy Problem with Dirichlet Boundary Data.- 3. The Solution to the Nonhomogeneous Cauchy Problem.- 4. Examples.- 5. Homogeneous Problems.- XVII. The Wave Equation.- 1. Introduction, the Cauchy Problem.- 2. Solutions in 1, 2 and 3 Dimensions.- 3. The Solution to the Nonhomogeneous Cauchy Problem.- 4. Examples.- Appendix I The Spectral Resolution of an Unbounded Self-Adjoint Operator.- 1. Unbounded Linear Operators.- 2. The Graph of an Operator.- 3. Symmetric and Self-Adjoint Operators.- 4. The Spectral Resolution of an Unbounded Self-Adjoint Operator.- Appendix II The Derivation of the Heat, Wave and Lapace Equations.- 1. The Heat Equation.- 2. Boundary Conditions.- 3. The Wave Equation.- 4. Boundary Conditions.- 5. Laplace’s Equation.

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