Introduction to Orthogonal Polynomials

By: THEODORE S CHIHARA

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Paperback | 17 February 2011

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272 Pages

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Assuming no further prerequisites than a first undergraduate course in real analysis, this concise introduction covers general elementary theory related to orthogonal polynomials. It includes necessary background material of the type not usually found in the standard mathematics curriculum. Suitable for advanced undergraduate and graduate courses, it is also appropriate for independent study.

Topics include the representation theorem and distribution functions, continued fractions and chain sequences, the recurrence formula and properties of orthogonal polynomials, special functions, and some specific systems of orthogonal polynomials. Numerous examples and exercises, an extensive bibliography, and a table of recurrence formulas supplement the text.

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Non-Fiction Mathematics Calculus & Mathematical Analysis Calculus Applied Mathematics

Preface	p. vii
Elementary Theory of Orthogonal Polynomials	p. 1
Introduction	p. 1
The moment functional and orthogonality	p. 6
Existence of OPS	p. 11
The fundamental recurrence formula	p. 18
Zeros	p. 26
Gauss quadrature	p. 31
Kernel polynomials	p. 35
Symmetric moment functionals	p. 40
Certain related recurrence relations	p. 45
The Representation Theorem and Distribution Functions	p. 51
Introduction	p. 51
Some preliminary theorems	p. 52
The representation theorem	p. 56
Spectral points and zeros of orthogonal polynomials	p. 59
Determinacy of L in the bounded case	p. 63
The classical moment problems	p. 71
Continued Fractions and Chain Sequences	p. 77
Basic concepts	p. 77
The fundamental recurrence formulas	p. 80
A convergence theorem	p. 82
Jacobi fractions and orthogonal polynomials	p. 85
Chain sequences	p. 91
Additional results on chain sequences	p. 100
The Recurrence Formula and Properties of Orthogonal Polynomials	p. 107
Introduction	p. 107
Chain sequences and orthogonal polynomials	p. 108
Some spectral analysis	p. 113
OPS whose zeros are dense in intervals	p. 120
Preliminaries to Krein's theorem	p. 128
Krein's theorem	p. 133
Special Functions	p. 142
General remarks	p. 142
The classical orthogonal polynomials	p. 142
The Hahn class of orthogonal polynomials	p. 159
The Meixner class of orthogonal polynomials	p. 163
Other classes of orthogonal polynomials	p. 166
Some Specific Systems of Orthogonal Polynomials	p. 170
The Charlier polynomials	p. 170
The Stieltjes-Wigert polynomials	p. 172
The Meixner polynomials	p. 175
The Bessel polynomials	p. 181
The Pollaczek polynomials	p. 184
Modified Lommel polynomials	p. 187
Tricomi-Carlitz polynomials	p. 190
OPS related to Bernoulli numbers	p. 191
OPS related to Jacobi elliptic functions	p. 193
The q-polynomials of Al-Salam and Carlitz	p. 195
Wall polynomials	p. 198
Associated Legendre polynomials	p. 201
Miscellaneous OPS	p. 203
Notes	p. 209
Appendix: Table of Recurrence Formulas	p. 215
List of Frequently Used Symbols	p. 223
Bibliography	p. 225
Index	p. 243
Table of Contents provided by Ingram. All Rights Reserved.

Introduction to Orthogonal Polynomials

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