Approximation Theory and Harmonic Analysis on Spheres and Balls 1 Spherical Harmonics 2 Convolution and Spherical Harmonic Expansion 3 Littlewood Paley Theory and Multiplier Theorem 4 Approximation on the Sphere 5 Weighted Polynomial Inequalities 6 Cubature Formulas on Spheres 7 Harmonic Analysis Associated to Reflection Groups 8 Boundedness of Projection Operator and Cesàro Means 9 Projection Operators and Cesàro Means in L^p Spaces 10 Weighted Best Approximation by Polynomials 11 Harmonic Analysis on the Unit Ball 12 Polynomial Approximation on the Unit Ball 13 Harmonic Analysis on the Simplex 14 Applications A Distance Difference and Integral Formulas B Jacobi and Related Orthogonal Polynomials References Index Symbol Index. This monograph records progress in approximation theory and harmonic analysis on balls and spheres and presents contemporary material that will be useful to analysts in this area. While the first part of the book contains mainstream material on the subject the second and the third parts deal with more specialized topics such as analysis in weight spaces with reflection invariant weight functions and analysis on balls and simplexes. The last part of the book features several applications including cubature formulas distribution of points on the sphere and the reconstruction algorithm in computerized tomography. This book is directed at researchers and advanced graduate students in analysis. Mathematicians who are familiar with Fourier analysis and harmonic analysis will understand many of the concepts that appear in this manuscript: spherical harmonics the Hardy Littlewood maximal function the Marcinkiewicz multiplier theorem the Riesz transform and doubling weights are all familiar tools to researchers in this area.