Orthogonal Functions Revised English Edition2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

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Ⅰ．Expansion in Series of Orthogonal Functions and Preliminary Notions of Hilbert Space1

1．Square Integrable Functions1

2．Linearly Independent Functions2

3．Elementary Notions of Hilbert Space5

4．Linear Approximations to Functions10

5．Convergence in the Mean12

6．Expansion in Series of Orthogonal Functions18

7．Orthogonal Cartesian Systems of Hilbert Space26

8．Lp Integrability．The Holder-Riesz and the Minkowski Inequalities31

9．Generalized Convergence in the Mean of Order p34

Ⅱ．Expansions in Fourier Series39

1．Approximation in the Mean of a Function by a Trigonometric Polynomial of Order n39

2．Convergence in the Mean of the Fourier Series of a Square In-tegrable Function41

3．Continuous Functions:Sufficient Conditions for Pointwise Con-vergence49

4．Criteria for Pointwise Convergence55

5．Term by Term Integration of the Fourier Series:The Hardy-Littlewood Criterion for Pointwise Convergence78

6．Fejér(C,1)Summability of Fourier Series86

7．(C,k)Summability(k>0)of Fourier Series107

8．Poisson＇s Method of Summing Fourier Series114

9．The Fourier Integral120

10．Gibbs＇Phenomenon141

11．Inequalities for the Partial Sums of Fourier Series of a Function of Bounded Variation148

12．Applications of Fourier Series150

13．The Fourier Transform158

Ⅲ．Expansions in Series of Legendre Polynomials and Spherical Harmonics169

1．Legendre Polynomials169

2．Schlafli＇s Integral Formula175

3．Differential Equations of Legendre Polynomials175

4．Recurrence Formulas for Legendre Polynomials176

5．The Christoffel Formula of Summation179

6．Laplace＇s Integral Formula for Pn(x)180

7．Mehler＇s Formulas182

8．Zeros of the Legendre Polynomials:Bruns＇Inequalities186

9．The Complete Orthonormal System {[1/2(2n+1)]1/2Pn(x)}189

10．Stieltjes＇Bounds for Legendre Polynomials195

11．Series of Legendre Polynomials for Functions of Bounded Variation:Picone＇s and Jackson＇s Theorems202

12．Formulas and Series for Asymptotic Approximation of Legendre Polynomials208

13．Limits of Integrals:Singular Integrals216

14．Convergence of Series of Legendre Polynomials:Hobson＇s Theo-rem220

15．Series of Stieltjes-Neumann240

16．Series of Legendre Polynomials for a Finite Interval244

17．Ferrers＇Functions Associated with Legendre Functions246

18．Harmonic Polynomials and Spherical Harmonics253

19．Integral Properties of Spherical Harmonics and the Addition Theorem for Legendre Polynomials263

20．Completeness of Spherical Harmonics with Respect to Square Integrable Functions270

21．Laplace Series for an Integrable Function272

22．Criterion for Pointwise Convergence of Laplace Series273

23．(C,k)Summation of Laplace Series275

24．Poisson Summation of Laplace Series287

25．The Poisson Sum of Legendre Series291

Ⅳ．Expansions in Laguerre and Hermite Series295

1．Laguerre Polynomials295

2．Hermite Polynomials and Tchebychef Orthogonal Polynomials303

3．Zeros of the Hermite and Laguerre Polynomials312

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 Functions349

8．Bessel＇s Equality for Infinite Intervals355

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 Convergence371

11．Series of Laguerre Polynomials382

APPENDIX386

BIBLIOGRAPHY399

INDEX409