Applied Analysis

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin', van Gu!ik. 'g The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma. coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

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.