Orthogonal Functions (Dover Phoenix Editions Series) Book

I.	Expansion in Series of Orthogonal Functions and Preliminary Notions of Hilbert Space	1
1.	Square Integrable Functions	1
2.	Linearly Independent Functions	2
3.	Elementary Notions of Hilbert Space	5
4.	Linear Approximations to Functions	10
5.	Convergence in the Mean	12
6.	Expansion in Series of Orthogonal Functions	18
7.	Orthogonal Cartesian Systems of Hilbert Space	26
8.	L[superscript p] Integrability. The Holder-Riesz and the Minkowski Inequalities	31
9.	Generalized Convergence in the Mean of Order p	34
II.	Expansions in Fourier Series	39
1.	Approximation in the Mean of a Function by a Trigonometric Polynomial of Order n	39
2.	Convergence in the Mean of the Fourier Series of a Square Integrable Function	41
3.	Continuous Functions: Sufficient Conditions for Pointwise Convergence	49
4.	Criteria for Pointwise Convergence	55
5.	Term by Term Integration of the Fourier Series: The Hardy-Littlewood Criterion for Pointwise Convergence	78
6.	Fejer (C, 1) Summability of Fourier Series	86
7.	(C, k) Summability (k [greater than sign] 0) of Fourier Series	107
8.	Poisson's Method of Summing Fourier Series	114
9.	The Fourier Integral	120
10.	Gibbs' Phenomenon	141
11.	Inequalities for the Partial Sums of Fourier Series of a Function of Bounded Variation	148
12.	Applications of Fourier Series	150
13.	The Fourier Transform	158
III.	Expansions in Series of Legendre Polynomials and Spherical Harmonics	169
1.	Legendre Polynomials	169
2.	Schlafli's Integral Formula	175
3.	Differential Equations of Legendre Polynomials	175
4.	Recurrence Formulas for Legendre Polynomials	176
5.	The Christoffel Formula of Summation	179
6.	Laplace's Integral Formula for P[subscript n](x)	180
7.	Mehler's Formulas	182
8.	Zeros of the Legendre Polynomials: Bruns' Inequalities	186
9.	The Complete Orthonormal System {[1/2(2n+1) superscript 1/2]P[subscript n](x)@{	189
10.	Stieltjes' Bounds for Legendre Polynomials	195
11.	Series of Legendre Polynomials for Functions of Bounded Variation: Picone's and Jackson's Theorems	202
12.	Formulas and Series for Asymptotic Approximation of Legendre Polynomials	208
13.	Limits of Integrals: Singular Integrals	216
14.	Convergence of Series of Legendre Polynomials: Hobson's Theorem	220
15.	Series of Stieltjes-Neumann	240
16.	Series of Legendre Polynomials for a Finite Interval	244
17.	Ferrers' Functions Associated with Legendre Functions	246
18.	Harmonic Polynomials and Spherical Harmonics	253
19.	Integral Properties of Spherical Harmonics and the Addition Theorem for Legendre Polynomials	263
20.	Completeness of Spherical Harmonics with Respect to Square Integrable Functions	270
21.	Laplace Series for an Integrable Function	272
22.	Criterion for Pointwise Convergence of Laplace Series	273
23.	(C, k) Summation of Laplace Series	275
24.	Poisson Summation of Laplace Series	287
25.	The Poisson Sum of Legendre Series	291
IV.	Expansions in Laguerre and Hermite Series	295
1.	Laguerre Polynomials	295
2.	Hermite Polynomials and Tchebychef Orthogonal Polynomials	303
3.	Zeros of the Hermite and Laguerre Polynomials	312
4.	Relations between the Polynomials L[superscript (alpha) subscript n](x) and H[subscript n](x)	318
5.	Formulas for Asymptotic Approximation of the Polynomials H[subscript n](x)	320
6.	Formulas for Asymptotic Approximation of the Polynomials L[superscript (alpha) subscript n](x)	333
7.	Completeness of the Polynomials L[superscript (alpha) subscript n](x) and H[subscript n](x) with Respect to Square Integrable Functions	349
8.	Bessel's Equality for Infinite Intervals	355
9.	Criteria for Uniform Convergence of the Series of Polynomials L[superscript (alpha) subscript n](x) and H[subscript n](x)	361
10.	Pointwise Convergence of the Series of Type h and Uspensky's Criterion for Convergence	371
11.	Series of Laguerre Polynomials	382
	Appendix	386
	Bibliography	399
	Index	409