Series Preface
Preface
1 Linear Spaces
1.1 Linear spaces
1.2 Normed spaces
1.2.1 Convergence
1.2.2 Banach spaces
1.2.3 Completion of normed spaces
1.3 Inner product spaces
1.3.1 Hilbert spaces
1.3.2 Orthogonality
1.4 Spaces of continuously differentiable functions
1.4.1 HSlder spaces
1.5 Lp spaces
1.6 Compact sets
2 Linear Operators on Normed Spaces
2.1 Operators
2.2 Continuous linear operators
2.2.1 C(V, W) as a Banach space
2.3 The geometric series theorem and its variants
2.3.1 A generalization
2.3.2 A perturbation result
2.4 Some more results on linear operators
2.4.1 An extension theorem
2.4.2 Open mapping theorem
2.4.3 Principle of uniform boundedness
2.4.4 Convergence of numerical quadratures
2.5 Linear functionals
2.5.1 An extension theorem for linear functionals
2.5.2 The Riesz representation theorem
2.6 Adjoint operators
2.7 Weak convergence and weak compactness
2.8 Compact linear operators
2.8.1 Compact integral operators on C(D)
2.8.2 Properties of compact operators
2.8.3 Integral operators on L2(a,b)
2.8.4 The Fredholm alternative theorem
2.8.5 Additional results on Fredholm integral equations
2.9 The resolvent operator
2.9.1 R(r)as a holomorphic function
3 Approximation Theory
3.1 Approximation of continuous functions by polynomials
3.2 Interpolation theory
3.2.1 Lagrange polynomial interpolation
3.2.2 Hermite polynomial interpolation
3.2.3 Piecewise polynomial interpolation
3.2.4 Trigonometric interpolation
3.3 Best approximation
3.3.1 Convexity, lower semicontinuity
3.3.2 Some abstract existence results
3.3.3 Existence of best approximation
3.3.4 Uniqueness of best approximation
3.4 Best approximations in inner product spaces, projection or closed convex sets
3.5 Orthogonal polynomials
3.6 Projection operators
3.7 Uniform error bounds
3.7.1 Uniform error bounds for L2-approximations
3.7.2 L2-approximations using polynomials
3.7.3 Interpolatory projections and their convergence
4 Fourier Analysis and Wavelets
4.1 Fourier series
4.2 Fourier transform
4.3 Discrete Fourier transform
4.4 Haar wavelets
4.5 Multiresolution analysis
5 Nonlinear Equations and Their Solution by Iteration
6 Finite Difference Method
7 Sobolev Spaces
8 Weak Formulations of Elliptic Boundary Value Problems
9 The Galerkin Method and Its Variants
10 Finite Element Analysis
11 Elliptic Variational Inequalities and Their Numerical Approximations
12 Numerical Solution of Fredholm Integral Equations of the Second Kind
13 Boundary Integral Equations
14 Multivariable Polynomial Approximations
References
Index