Introduction to the theory of weighted polynomial approximation / H.N. Mhaskar.

Format:

Book

Author/Creator:

Mhaskar, H. N. (Hrushikesh Narhar), 1956-

Series:

Series in approximations and decompositions ; vol. 7.

Series in approximations and decompositions ; vol. 7

Language:

English

Subjects (All):

Approximation theory.

Orthogonal polynomials.

Physical Description:

1 online resource (398 p.)

Place of Publication:

Singapore ; River Edge, NJ : World Scientific, c1996.

Language Note:

English

Summary:

In this book, we have attempted to explain a variety of different techniques and ideas which have contributed to this subject in its course of successive refinements during the last 25 years. There are other books and surveys reviewing the ideas from the perspective of either potential theory or orthogonal polynomials. The main thrust of this book is to introduce the subject from an approximation theory point of view. Thus, the main motivation is to study analogues of results from classical trigonometric approximation theory, introducing other ideas as needed. It is not our objective to survey the most recent results, but merely to introduce to the readers the thought processes and ideas as they are developed. This book is intended to be self-contained, although the reader is expected to be familiar with rudimentary real and complex analysis. It will also help to have studied elementary trigonometric approximation theory, and have some exposure to orthogonal polynomials.

Contents:

1. Orthogonal polynomials. 1.1. Elementary facts. 1.2. Interpolation and quadrature. 1.3. Extremal properties. 1.4. Some estimates

2. Approximation on [-1, 1]. 2.1. Inequalities for trigonometric polynomials. 2.2. Favard-type estimates. 2.3. A K-functional and the degree of approximation

3. Freud polynomials. 3.1. The weight function and orthogonal polynomials. 3.2. Christoffel functions. 3.3. An alternative approach. 3.4. Polynomial inequalities

4. Degree of approximation. 4.1. Favard-type estimates. 4.2. Direct and converse theorems. 4.3. The Fourier transform

5. The K-functional. 5.1. A smooth weight function. 5.2. A modulus of smoothness. 5.3. Alternative expressions

6. Potential theory. 6.1. Where does the sup norm live? 6.2. Where does the L[symbol]-norm live? 6.3. Some extremal polynomials. 6.4. The prototypical Freud weights

7. Approximation of entire functions. 7.1. Classical results. 7.2. Functions with finite order and type. 7.3. Functions of finite exponential type

8. Freud polynomials II. 8.1. The interior bounds. 8.2. The Lubinsky entire function. 8.3. The largest zero

9. Processes of approximation. 9.1. Functions of bounded variation. 9.2. Interpolation and quadrature. 9.3. Strip of convergence

10. A density theorem. 10.1. The contour integral approach. 10.2. Discretization of the potential. 10.3. The strong asymptotics

11. Applications. 11.1. Weighted potentials. 11.2. Gaussian networks. 11.3. Wavelets.

Notes:

Bibliographic Level Mode of Issuance: Monograph

Includes bibliographical references (p. 355-379) and index.

ISBN:

9781299280939

1299280935

9789812832139

9812832130

OCLC:

831661863