Author | Lebedev, V. I. author |
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Title | An Introduction to Functional Analysis in Computational Mathematics [electronic resource] / by V. I. Lebedev |
Imprint | Boston, MA : Birkhรคuser Boston, 1997 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-4128-7 |
Descript | XII, 256 p. online resource |
1. Functional Spaces and Problems in the Theory of Approximation -- 1. Metric Spaces -- 2. Compact Sets in Metric Spaces -- 3. Statement of the Main Extremal Problems in the Theory of Approximation. Main Characteristics of the Best Approximations -- 4. The Contraction Mapping Principle -- 5. Linear Spaces -- 6. Normed and Banach Spaces -- 7. Spaces with an Inner Product. Hilbert Spaces -- 8. Problems on the Best Approximation. Orthogonal Expansions and Fourier Series in a Hilbert Space -- 9. Some Extremal Problems in Normed and Hilbert Spaces -- 10. Polynomials the Least Deviating from Zero. Chebyshev Polynomials and Their Properties -- 11. Some Extremal Polynomials -- 2. Linear Operators and Functionals -- 1. Linear Operators in Banach Spaces -- 2. Spaces of Linear Operators -- 3. Inverse Operators. Linear Operator Equations. Condition Measure of Operator -- 4. Spectrum and Spectral Radius of Operator. Convergence Conditions for the Neumann Series. Perturbations Theorem -- 5. Uniform Boundedness Principle -- 6. Linear Functionals and Adjoint Space -- 7. The Riesz Theorem. The Hahn-Banach Theorem. Optimization Problem for Quadrature Formulas. The Duality Principle -- 8. Adjoint, Selfadjoint, Symmetric Operators -- 9. Eigenvalues and Eigenelements of Selfadjoint and Symmetric Operators -- 10. Quadrature Functionals with Positive Definite Symmetric or Symmetrizable Operator and Generalized Solutions of Operator Equations -- 11. Variational Methods for the Minimization of Quadrature Functionals -- 12. Variational Equations. The Vishik-Lax-Milgram Theorem -- 13. Compact (Completely Continuous) Operators in Hilbert Space -- 14. The Sobolev Spaces. Embedding Theorems -- 15. Generalized Solution of the Dirichlet Problem for Elliptic Equations of the Second Order -- 3. Iteration Methods for the Solution of Operator Equations -- 1. General Theory of Iteration Methods -- 2. On the Existence of Convergent Iteration Methods and Their Optimization -- 3. The Chebyshev One-Step (Binomial) Iteration Methods -- 4. The Chebyshev Two-Step (Trinomial) Iteration Method -- 5. The Chebyshev Iteration Methods for Equations with Symmetrized Operators -- 6. Block Chebyshev Method -- 7. The Descent Methods -- 8. Differentiation and Integration of Nonlinear Operators. The Newton Method -- 9. Partial Eigenvalue Problem -- 10. Successive Approximation Method for Inverse Operator -- 11. Stability and Optimization of Explicit Difference Schemes for Stiff Differential Equations