Chapter 1 Preliminaries 1
1.1 Basic concepts 1
1.1.1 Definition of ？,？and？ 1
Contents 1
1.1.2 L2(Ωn)(n≥2) 4
1.1.3 The case n＝2 6
1.1.4 Zonal harmonics 7
1.1.5 Representation for spherical harmonics 14
1.1.6 Laplace-Beltrami Operator 16
1.1.7 The convolution for functions on sphere 17
1.2.1 Rodrigues formula 21
1.2 Gegenbauer and Jacobi polynomials 21
1.2.2 Funk-Hecke formula 24
1.2.3 Laplace representation 26
1.2.4 Generating formulas 27
1.2.5 The leading coefficient of？ 31
1.2.6 Differential equations for？ 31
1.2.7 Jacobi Polynomials 32
1.3 Jacobi polynomials with complex indices 34
Chapter 2 Fourier-Laplace Series 43
2.1 Introduction 43
2.2 Convergence,Lebesgue constant 45
2.3 Cesàro means(Early results) 48
2.4 Translation operator and mean operator 56
2.5 Maximal translation operator 63
2.5.1 The proof of Theorem 2.5.1 64
2.5.2 Proof of Theorem 2.5.2 71
2.6 Projection operators 75
Chapter 3 Equiconvergent Operators of Cesàro Means 85
3.1 Definition 85
3.2 Localization 95
3.2.1 The case δ≥n－2 95
-1＜δ＜n－2 97
3.2.2 The necessity of antipole conditions when 97
3.2.3 Antipole conditions when？－1＜δ＜n－2 100
3.2.4 Corollary of Theorems 3.2.5 and 3.2.6 105
3.3 Pointwise convergence 106
3.3.1 Equivalent conditionsfor convergence 106
3.3.2 Tests for convergence 108
3.3.3 A test of Salem type 113
3.4 Maximal operatorE？ and a.e.convergence 116
3.5 Application to linear summability 129
3.5.1 Introduction 129
3.5.2 Auxiliary lemmas 131
3.5.3 Convergence everywhere 142
3.5.4 Convergence at Lebesgue points 150
Chapter 4 Constructive Properties of Spherical Functions 161
4.1 Best approximation operator 161
4.2 Pointwise Derivatives 162
4.2.1 Preliminary 163
4.2.2 Estimate for the tangent gradients 165
4.2.3 Estimate for the normal gradient of harmonicpolynomials 168
4.3 Fractional derivative and integral 170
4.4.1 Definitinitions 172
4.4 Fractional integrals of variable order 172
4.4.2 Propertiesof Poisson integrals on the sphere 174
4 4.3 Proof of Theorem 4.4.1 180
4.5 Modulus of continuity 182
4.6 Derivatives and finite differences 188
Chapter 5 Jackson Type Theorems 193
5.1 Jackson inequality and K-functional 193
5.1.1 Estimates for ultraspherical polynomials 194
5.1.2 Estimates for the best approximation 212
5.1.3 Estimate for derivative of polynomials 214
5.1.4 Proofof Theorems 5.1.1 and 5.1.2 215
5.2 Difference*△？and space H？ 216
Chapter 6 Approximation by Linear Means 229
6.1 Almost everywhere approximation 229
6.1.1 Introduction 229
6.1 2 Approximation by Riesz means on sets of 231
full measure 231
6.1.3 Approximation by partial sums on sets of 236
full measure 236
6.1.4 Strong approximation by Cesàro Means 241
6.2 Approximation in norm 254
6.2.1 Riesz means and Peetre K-mmoduli 254
6.2.2 Riesz means and the best approximation 257
6.2.3 Riesz means with critical index 259
6.2.4 Riesz means and Cesàro means 263
6.3 The de la Vallée Poussin Means 266
6.3.1 Convergence and approximation in norm 268
6.3.2 Pointwise convergence and approximation 270
6.3.3 Weak type inequalities for the best approximation 273
6.3.4 Characterization through a classical modulus ofsmoothness in C 275
6.3.5 Approximation for zonal functions 280
References 285
Index 299