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Harmonic Analysis and Approximation on the Unit Sphere 球面上的调和分析与逼近PDF|Epub|txt|kindle电子书版本下载
- Wang Kunyang，Li Luoqing著 著
- 出版社： 北京：科学出版社
- ISBN：7030083660
- 出版时间：2000
- 标注页数：300页
- 文件大小：6MB
- 文件页数：309页
- 主题词：
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图书目录
Chapter 1 Preliminaries1
1.1 Basic concepts1
1.1.1 Definition of ？,？and？1
Contents1
1.1.2 L2(Ωn)(n≥2)4
1.1.3 The case n＝26
1.1.4 Zonal harmonics7
1.1.5 Representation for spherical harmonics14
1.1.6 Laplace-Beltrami Operator16
1.1.7 The convolution for functions on sphere17
1.2.1 Rodrigues formula21
1.2 Gegenbauer and Jacobi polynomials21
1.2.2 Funk-Hecke formula24
1.2.3 Laplace representation26
1.2.4 Generating formulas27
1.2.5 The leading coefficient of？31
1.2.6 Differential equations for？31
1.2.7 Jacobi Polynomials32
1.3 Jacobi polynomials with complex indices34
Chapter 2 Fourier-Laplace Series43
2.1 Introduction43
2.2 Convergence,Lebesgue constant45
2.3 Cesàro means(Early results)48
2.4 Translation operator and mean operator56
2.5 Maximal translation operator63
2.5.1 The proof of Theorem 2.5.164
2.5.2 Proof of Theorem 2.5.271
2.6 Projection operators75
Chapter 3 Equiconvergent Operators of Cesàro Means85
3.1 Definition85
3.2 Localization95
3.2.1 The case δ≥n－295
-1＜δ＜n－297
3.2.2 The necessity of antipole conditions when97
3.2.3 Antipole conditions when？－1＜δ＜n－2100
3.2.4 Corollary of Theorems 3.2.5 and 3.2.6105
3.3 Pointwise convergence106
3.3.1 Equivalent conditionsfor convergence106
3.3.2 Tests for convergence108
3.3.3 A test of Salem type113
3.4 Maximal operatorE？ and a.e.convergence116
3.5 Application to linear summability129
3.5.1 Introduction129
3.5.2 Auxiliary lemmas131
3.5.3 Convergence everywhere142
3.5.4 Convergence at Lebesgue points150
Chapter 4 Constructive Properties of Spherical Functions161
4.1 Best approximation operator161
4.2 Pointwise Derivatives162
4.2.1 Preliminary163
4.2.2 Estimate for the tangent gradients165
4.2.3 Estimate for the normal gradient of harmonicpolynomials168
4.3 Fractional derivative and integral170
4.4.1 Definitinitions172
4.4 Fractional integrals of variable order172
4.4.2 Propertiesof Poisson integrals on the sphere174
4 4.3 Proof of Theorem 4.4.1180
4.5 Modulus of continuity182
4.6 Derivatives and finite differences188
Chapter 5 Jackson Type Theorems193
5.1 Jackson inequality and K-functional193
5.1.1 Estimates for ultraspherical polynomials194
5.1.2 Estimates for the best approximation212
5.1.3 Estimate for derivative of polynomials214
5.1.4 Proofof Theorems 5.1.1 and 5.1.2215
5.2 Difference*△？and space H？216
Chapter 6 Approximation by Linear Means229
6.1 Almost everywhere approximation229
6.1.1 Introduction229
6.1 2 Approximation by Riesz means on sets of231
full measure231
6.1.3 Approximation by partial sums on sets of236
full measure236
6.1.4 Strong approximation by Cesàro Means241
6.2 Approximation in norm254
6.2.1 Riesz means and Peetre K-mmoduli254
6.2.2 Riesz means and the best approximation257
6.2.3 Riesz means with critical index259
6.2.4 Riesz means and Cesàro means263
6.3 The de la Vallée Poussin Means266
6.3.1 Convergence and approximation in norm268
6.3.2 Pointwise convergence and approximation270
6.3.3 Weak type inequalities for the best approximation273
6.3.4 Characterization through a classical modulus ofsmoothness in C275
6.3.5 Approximation for zonal functions280
References285
Index299