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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．Lp 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 In-tegrable Function 41

3．Continuous Functions:Sufficient Conditions for Pointwise Con-vergence 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．Fejér(C,1)Summability of Fourier Series 86

7．(C,k)Summability(k>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 Pn(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)]1/2Pn(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 Theo-rem 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 Ln(a)(x)and Hn(x) 318

5．Formulas for Asymptotic Approximation of the Polynomials Hn(x) 320

6．Formulas for Asymptotic Approximation of the Polynomials Ln(a)(x) 333

7．Completeness of the Polynomials Ln(a)(x)and Hn(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 Ln(a)(x)and Hn(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