The aim of the book "Harmonic Analysis and Approximation on the Unit Sphere" is:(1) to give a basic introduction to the spherical harmonic expansion i.e.the Fourier-Laplace series of functions on the unit sphere of multidimensional Euclidian spaces:(2)to present the main research results obtained in this area by the authors and their CO-authors during the period from 199l to l999:(3) to provide a basic self-contained reading material for the graduate students. This book can serve as a specialty textbook for the graduate students in the real function theory group of mathematical department of universities.
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目录
Chapter 1 Preliminaries 1.1 Basic concepts 1.1.1 Definition of Hk的n次方，Ak的n次方and Hk的n次方 1.1.2 L的2次方(Ωn)(n≥2) 1.1.3 The case n=2 1.1.4 Zonal hamonics 1.1.5 Representation for spherical harmonics 1.1.6 Laplace-Beltrami Operator 1.1.7 The convolution for functions on sphere 1.2 Gegenbauer and Jacobi polynomials 1.2.1 Rodrigues' formula 1.2.2 Funk—Hecke formula 1.2.3 Laplace representation 1.2.4 Generating formulas 1.2.5 The leading coefficient of Pk的n次方 1.2.6 Differential equations for Pk的n次方 1.2.7 Jacobi Polynomials 1.3 Jacobi polynomials with complex indices Chapter 2 Fourier-Laplace Series 2.1 Introduction 2.2 Convergence，Lebesgue constant 2.3 Cesàro means(Early result) 2.4 Translation operator and mean operator 2.5 Maximal translation operator 2.5.1 The proof of Theorem 2.5.1 2.5.2 Proof of Theorem 2.5.2 2.6 Proiection operators Chapter 3 Equiconvergent Operators of Cesàro Means 3.1 Definition 3.2 Localization 3.2.1 The caseδ≥n一2 3.2.2 The necessity of "antipole conditions" when -1<δ3.2.3 Antipole conditions when n-2/2-1<δ3.2.4 Corollary of Theorems 3.2.5 and 3.2.6 3.3 Pointwise convergence 3.3.1 Equivalent conditions for convergence 3.3.2 Tests for convergence 3.3.3 A test of Salem type 3.4 Maximal operator E*的a次方and a．e．convergence 3.5 Application to linear summability 3.5.1 Introduction 3.5.2 Auxiliary lemmas 3.5.3 Convergence everywhere 3.5.4 Convergence at Lebesgue points Chapter 4 Constructive Properties of Spherical Functions 4.1 Best approximation operator 4.2 Pointwise Derivatives 4.2.1 Preliminary 4.2.2 Estimate for the tangent gradients 4.2.3 Estimate for the normal gradient of harmonic polynomials 4.3 Fractional derivative and integral 4.4 Fractional integrals of variable order 4.4.1 Definitions 4.4.2 Properties of Poisson integrals on the sphere 4.4.3 Proof of Theorem 4.4.1 4.5 Modulus of continuity 4.6 Derivatives and finite differences Chapter 5 Jackson Type Theorems 5.1 Jackson inequality and K-functional 5.1.1 Estimates for ultraspherical polynomials 5.1.2 Estimates for the best approximation 5.1.3 Estimate for derivative of polynonfials 5.1.4 Proof of Theorems 5.1.l and 5.1.2 5.2 Difference*△t的k次方and space H的r次方 Chapter 6 Approximation by Linear Means 6.1 Almost everywhere approximation 6.1.1 Introduction 6.1.2 Approximation by Riesz nleans on sets of full measure 6.1.3 Approximation by partial sums on sets of full measure 6.1.4 Strong approximation by Cesàro Means 6.2 Approximation in norm 6.2.1 Riesz means and Peetre K-moduli 6.2.2 Riesz means and the best approximation 6.2.3 Riesz means with critical index 6.2.4 Riesz means and Cesàro means 6.3 The de la Vallée Poussin Means 6.3.1 Convergence and approximation in norm 6.3.2 Pointwise convergence and approximation 6.3.3 Weak type inequalities for the best approximation 6.3.4 Characterization through a classical modulus of smoothness in C 6.3.5 Approximation for zonal functions Referenees Index
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