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Approximation theory and harmonic analysis on spheres and balls. (English) Zbl 1275.42001

Springer Monographs in Mathematics. New York, NY: Springer (ISBN 978-1-4614-6659-8/hbk; 978-1-4614-6660-4/ebook). xviii, 440 p. (2013).
The book under review is the most detailed monograph on harmonic analysis, approximation and their applications in the spherical setting.
The book is naturally divided into four parts. The first one is devoted to approximation and harmonic analysis on the sphere. The first six chapters correspond to this part. The first one is of preliminary nature, while the fourth one is about approximation. More details can be found in a recent book [K. Atkinson and W. Han, Spherical harmonics and approximation on the unit sphere: An introduction. Berlin: Springer (2012; Zbl 1254.41015)]. The other two chapters present the setting of modern harmonic analysis to the unit sphere. Here are these four chapters: Chapter 1 Spherical Harmonics. Chapter 2 Convolution Operator and Spherical Harmonic Expansion. Chapter 3 Littlewood-Paley Theory and the Multiplier Theorem. Chapter 4 Approximation on the Sphere. In the next Chapter 5 Weighted Polynomial Inequalities, the authors start to deal with weighted approximation and, in particular, a recent proof of a conjecture on spherical design due to A. Bondarenko, D. Radchenko and M. Viazovska is contained in it. In Chapter 6 Cubature Formulas on Spheres, an introduction to cubature formulas, which are necessary for discretizing integrals, is given.
The second part, strongly related to the fourth chapter, consists of the next four chapters. In these chapters, analysis in weighted spaces on the sphere is provided. They are: Chapter 7 Harmonic Analysis Associated with Reflection Groups. Chapter 8 Boundedness of Projection Operators and Cesàro Means. Chapter 9 Projection Operators and Cesàro Means in \(L^p\) Spaces. Chapter 10 Weighted Best Approximation by Polynomials.
The third part deals with analysis on the unit ball in the two chapters Chapter 11 Harmonic Analysis on the Unit Ball and Chapter 12 Polynomial Approximation on the Unit Ball, while Chapter 13 Harmonic Analysis on the Simplex is concerned with similar problems on the simplex. I am not aware of any other book where these problems are considered in such a systematic way.
The fourth part consists of one chapter Chapter 14 Applications, in which the above studied theories are applied to various topics. Of course, a separate book can be written on this, but the authors reasonably chose certain applications, to their taste, which well illustrate the previous theoretical matter.
There are two appendices in the book, A: Distance, Difference and Integral Formulas and B: Jacobi and Related Orthogonal Polynomials.
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.
There are minor remarks that probably express the reviewer’s taste. More comparison with the classical trigonometric case is desirable. Though the list of references contains 197 items, there is a feeling that some sources could be added to it, and not only for making it longer than 200 items. These are, for example, the book by S. B. Topuria [Fourier-Laplace series, Tbilisi (1987) (Russian)], and a survey paper by L. V. Zhizhiashvili and S. B. Topuria, [Itogi Nauki Tekh., Ser. Mat. Anal. 15, 83–130 (1977; Zbl 0416.42015) (Russian); Engl. transl. in J. Sov. Math. 12, 682–714 (1979; Zbl 0471.42001)]. But the most amazing is the absence of a very important paper by E. Kogbetliantz [Journ. de Math. (9) 3, 107–187 (1924; JFM 50.0207.05)]. To this end, we mention that one of the basic results of this paper is given explicitly on p. 51 but referred to [E. R. Liflyand, Acta Sci. Math. 64, No. 1–2, 215–222 (1998; Zbl 0928.42008)] and [C. D. Sogge, Duke Math. J. 53, 43–65 (1986; Zbl 0636.42018)].

MSC:

42-02 Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces
41-02 Research exposition (monographs, survey articles) pertaining to approximations and expansions
42A10 Trigonometric approximation
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
43A90 Harmonic analysis and spherical functions
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