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Preface | |
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Distribution Theory and Green's Functions | |
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Generalised Functions | |
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The Delta Function | |
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Basic Distribution Theory | |
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Operations on Distributions | |
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Convergence of Distributions | |
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Further Developments | |
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Fourier Series and the Poisson Sum Formula | |
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Summary and References | |
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Problems | |
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Differential Equations and Green's Functions | |
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The Integral of a Distribution | |
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Linear Differential Equations | |
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Fundamental Solutions of Differential Equations | |
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Green's Functions | |
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Applications of Green's Functions | |
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Summary and References | |
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Problems | |
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Fourier Transforms and Partial Differential Equations | |
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The Classical Fourier Transform | |
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Distributions of Slow Growth | |
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Generalised Fourier Transforms | |
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Generalised Functions of Several Variables | |
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Green's Function for the Laplacian | |
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Green's Function for the Three-Dimensional Wave Equation | |
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Summary and References | |
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Problems | |
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Banach Spaces and Fixed Point Theorems | |
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Normed Spaces | |
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Vector Spaces | |
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Normed Spaces | |
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Convergence | |
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Open and Closed Sets | |
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Completeness | |
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Equivalent Norms | |
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Summary and References | |
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Problems | |
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The Contraction Mapping Theorem | |
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Operators on Vector Spaces | |
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The Contraction Mapping Theorem | |
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Application to Differential and Integral Equations | |
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Nonlinear Diffusive Equilibrium | |
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Nonlinear Diffusive Equilibrium in Three Dimensions | |
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Summary and References | |
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Problems | |
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Compactness and Schauder's Theorem | |
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Continuous Operators | |
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Brouwer's Theorem | |
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Compactness | |
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Relative Compactness | |
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Arzela's Theorem | |
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Schauder's Theorems | |
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Forced Nonlinear Oscillations | |
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Swirling Flow | |
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Summary and References | |
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Problems | |
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Operators in Hilbert Space | |
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Hilbert Space | |
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Inner Product Spaces | |
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Orthogonal Bases | |
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Orthogonal Expansions | |
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The Bessel, Parseval, and Riesz-Fischer Theorems | |
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Orthogonal Decomposition | |
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Functionals on Normed Spaces | |
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Functionals in Hilbert Space | |
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Weak Convergence | |
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Summary and References | |
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Problems | |
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The Theory of Operators | |
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Bounded Operators on Normed Spaces | |
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The Algebra of Bounded Operators | |
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Self-Adjoint Operators | |
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Eigenvalue Problems for Self-Adjoint Operators | |
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Compact Operators | |
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Summary and References | |
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Problems | |
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The Spectral Theorem | |
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The Spectral Theorem | |
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Sturm-Liouville Systems | |
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Partial Differential Equations | |
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The Fredholm Alternative | |
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Projection Operators | |
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Summary and References | |
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Problems | |
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Variational Methods | |
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Positive Operators | |
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Approximation to the First Eigenvalue | |
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The Rayleigh-Ritz Method for Eigenvalues | |
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The Theory of the Rayleigh-Ritz Method | |
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Inhomogeneous Equations | |
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Complementary Bounds | |
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Summary and References | |
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Problems | |
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Further Developments | |
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The Differential Calculus of Operators and its Applications | |
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The Frechet Derivative | |
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Higher Derivatives | |
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Maxima and Minima | |
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Linear Stability Theory | |
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Nonlinear Stability | |
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Bifurcation Theory | |
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Bifurcation and Stability | |
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Summary and References | |
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Distributional Hilbert Spaces | |
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The Space of Square-Integrable Distributions | |
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Sobolev Spaces | |
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Application to Partial Differential Equations | |
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Summary and References | |
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Appendices | |
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Sets and Mappings | |
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Sequences, Series, and Uniform Convergence | |
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Sup and Inf | |
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Countability | |
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Equivalence Relations | |
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Completion | |
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Sturm-Liouville Systems | |
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Fourier's Theorem | |
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Proofs of 9.24 and 9.25 | |
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Notes on the Problems | |
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Supplementary Problems | |
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Symbol Index | |
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References and Name Index | |
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Subject Index | |