Approximation Theory and Harmonic Analysis on Spheres and Balls

ISBN-13: 9781461466598 / Angielski / Twarda / 2013 / 440 str.

Feng Dai; Yuan Xu

Opis

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.

Recenzje

From the reviews:

"This research monograph is recommended to graduate students, mathematicians, physicists, and engineers who have an interest in analysis and approximation on the sphere, ball, and simplex. ... At the end of each chapter one finds useful 'notes and further results', where the authors present an account of the sources used for the developments in the chapter as well as comments on related results." (P. P. Petrushev, Mathematical Reviews, January, 2014)

"The book under review is the most detailed monograph on harmonic analysis, approximation and their applications in the spherical setting. ... This monograph in whole and its various parts can be used both by researchers and by lecturers, for information and ideas by the formers and as a matter for special courses for students by the latters." (Elijah Liflyand, zbMATH, Vol. 1275, 2014)

Spis treści

​​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.

O autorze

Feng Dai is currently a professor of mathematics at the University of Alberta, and Yuan Xu is currently a professor of mathematics at the University of Oregon.