Selected Topics in Approximation and Computation (International Series of Monographs on Computer Science)

✍ Scribed by Marek Kowalski, Christopher Sikorski, Frank Stenger

• Publisher: Oxford University Press
• Year: 1995
• Tongue: English
• Leaves: 366
• Category: Library

✦ Synopsis

Selected Topics in Approximation and Computation addresses the relationship between modern approximation theory and computational methods. The text is a combination of expositions of basic classical methods of approximation leading to popular splines and new explicit tools of computation, including Sinc methods, elliptic function methods, and positive operator approximation methods. It also provides an excellent summary of worst case analysis in information based complexity. It relates optimal computational methods with the theory of s-numbers and n-widths. It can serve as a text for senior-graduate courses in computer science and applied mathematics, and also as a reference for professionals.

✦ Table of Contents

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 Γ
3.3.1 Exercises
3.4 Sinc indefinite integral over Γ
3.4.1 Exercises
3.5 Sinc indefinite convolution over Γ
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 Csub(o)
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