
Selected Topics in Approximation and Computation
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Content
- Intro
- Contents
- 1 Classical Approximation
- 1.1 General results
- 1.1.1 Exercises
- 1.2 Approximation in unitary spaces
- 1.2.1 Computing the best approximation
- 1.2.2 Completeness of orthogonal systems
- 1.2.3 Examples of orthogonal systems
- 1.2.4 Remarks on convergence of Fourier series
- 1.2.5 Exercises
- 1.3 Uniform approximation
- 1.3.1 Chebyshev subspaces
- 1.3.2 Maximal functionals
- 1.3.3 The Remez algorithm
- 1.3.4 The Korovkin operators
- 1.3.5 Quality of polynomial approximations
- 1.3.6 Converse theorems in polynomial approximation
- 1.3.7 Projection operators
- 1.3.8 Exercises
- 1.4 Annotations
- 1.5 References
- 2 Splines
- 2.1 Polynomial splines
- 2.1.1 Exercises
- 2.2 B-splines
- 2.2.1 General spline interpolation
- 2.2.2 Exercises
- 2.3 General splines
- 2.3.1 Exercises
- 2.4 Annotations
- 2.5 References
- 3 Sinc Approximation
- 3.1 Basic definitions
- 3.1.1 Exercises
- 3.2 Interpolation and quadrature
- 3.2.1 Exercises
- 3.3 Approximation of derivatives on G
- 3.3.1 Exercises
- 3.4 Sinc indefinite integral over G
- 3.4.1 Exercises
- 3.5 Sinc indefinite convolution over G
- 3.5.1 Derivation and justification of procedure
- 3.5.2 Multidimensional indefinite convolutions
- 3.5.3 Two dimensional convolution
- 3.5.4 Exercises
- 3.6 Annotations
- 3.7 References
- 4 Explicit Sinc-Like Methods
- 4.1 Positive base approximation
- 4.1.1 Exercises
- 4.2 Approximation via elliptic functions
- 4.2.1 Exercises
- 4.3 Heaviside, filter, and delta functions
- 4.3.1 Heaviside function
- 4.3.2 The filter or characteristic function
- 4.3.3 The impulse or delta function
- 4.3.4 Exercises
- 4.4 Annotations
- 4.5 References
- 5 Moment Problems
- 5.1 Duality with approximation
- 5.1.1 Exercises
- 5.2 The moment problem in the space C[sub(o)](D)
- 5.3 Classical moment problems
- 5.3.1 Exercises
- 5.4 Density and determinateness
- 5.4.1 Exercises
- 5.5 A Sinc moment problem
- 5.5.1 Exercises
- 5.6 Multivariate orthogonal polynomials
- 5.6.1 Exercises
- 5.7 Annotations
- 5.8 References
- 6 n-Widths and s-Numbers
- 6.1 n-Widths
- 6.1.1 Relationships between n-widths
- 6.1.2 Algebraic versions of a[sub(n)] and c[sub(n)]
- 6.1.3 Exercises
- 6.2 s-Numbers
- 6.2.1 s-Numbers and singular values
- 6.2.2 Relationships between s-numbers
- 6.2.3 Exercises
- 6.3 Annotations
- 6.4 References
- 7 Optimal Approximation Methods
- 7.1 A general approximation problem
- 7.1.1 Radius of information-optimal algorithms
- 7.1.2 Exercises
- 7.2 Linear problems
- 7.2.1 Optimal information
- 7.2.2 Relations to n-widths
- 7.2.3 Exercises
- 7.3 Parallel versus sequential methods
- 7.3.1 Exercises
- 7.4 Linear and spline algorithms
- 7.4.1 Spline algorithms
- 7.4.2 Relations to linear Kolmogorov n-widths
- 7.4.3 Exercises
- 7.5 s-Numbers, minimal errors
- 7.5.1 Exercises
- 7.6 Optimal methods
- 7.6.1 Optimal complexity methods for linear problems
- 7.6.2 Exercises
- 7.7 Annotations
- 7.8 References
- 8 Applications
- 8.1 Sinc solution of Burgers' equation
- 8.2 Signal recovery
- 8.2.1 Formulation of the problem
- 8.2.2 Relations to n-widths
- 8.2.3 Algorithms and their errors
- 8.2.4 Asymptotics of minimal cost
- 8.2.5 Exercises
- 8.3 Bisection method
- 8.3.1 Formulation of the problem
- 8.3.2 Optimality theorem
- 8.3.3 Exercises
- 8.4 Annotations
- 8.5 References
- Index
- A
- B
- C
- D
- E
- F
- G
- H
- I
- J
- K
- L
- M
- N
- O
- P
- Q
- R
- S
- T
- U
- V
- W
- X
- Y
- Z
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