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Analysis of spherical symmetries in Euclidean spaces. (English) Zbl 0884.33001
Applied Mathematical Sciences. 129. New York, NY: Springer. viii, 223 p. (1998).
This is a book on special functions. How to define the various types of the classical special functions? In the very beginning (§2) of the book, the author states that a Legendre polynomial $$L_n(q;x)$$ of degree $$n$$ and of a $$q$$-dimensional variable $$x$$ is uniquely determined by $$L_n(q,Ax) = L_n(q,x)$$ with a certain unitary transform $$A$$ of the $$q$$-dimensional Euclidean space together with a suitable normalization.
This is the subject of the book: to determine and to study spherical harmonics in a theory of invariants of an orthogonal group. Everywhere the author emphasizes the structures that are common to all dimensions. The headings of the chapters can not give a complete impression of their extensive contents.
The first chapter culminates in the proof that there is only one system of invariant linear spaces of functions on the unit sphere, this is the system of spherical harmonics. The second chapter fills the general frame with explicit and concrete data as Legendre and Gegenbauer polynomials, further it formulates Maxwell’s theory of multipoles.
The third chapter is an extended version of the classical three-dimensional results on the Laplacian, the simplest orthogonal invariant differential operator. The next chapter extends the results obtained to an analysis on the complex unit sphere, e.g. it contains a complexified Funk-Hecke formula. The fifth chapter turns to the solutions of Helmholtz equation. Spherically symmetric spaces of its solutions are obtained by linear combinations of plane waves, thus introducing into the theory of Bessel functions differing from the usual treatments. The last two chapters, essentially an introduction into the main results of the theory of Fourier transforms and on Radon transforms, treat such integral transform with respect to the property ”preservation of spherical symmetry”. They gain new aspects of special topics in the theories of spherical harmonics, Bessel functions, and Fourier analysis.
Altogether, the monograph enables a thorough insight into the contemporary role of an old field such as that of special functions in modern constructive and concrete analysis.

##### MSC:
 33-02 Research exposition (monographs, survey articles) pertaining to special functions 33C10 Bessel and Airy functions, cylinder functions, $${}_0F_1$$ 33C55 Spherical harmonics 42Cxx Nontrigonometric harmonic analysis 44A12 Radon transform