By Claus Müller

This ebook offers a brand new and direct strategy into the theories of particular services with emphasis on round symmetry in Euclidean areas of ar bitrary dimensions. crucial elements may also be referred to as common a result of selected suggestions. The relevant subject is the presentation of round harmonics in a idea of invariants of the orthogonal crew. H. Weyl was once one of many first to show that round harmonics needs to be greater than a lucky bet to simplify numerical computations in mathematical physics. His opinion arose from his profession with quan tum mechanics and was once supported by means of many physicists. those principles are the major subject matter all through this treatise. whilst R. Richberg and that i begun this venture we have been stunned, how effortless and chic the overall concept may be. one of many highlights of this publication is the extension of the classical result of round harmonics into the advanced. this can be fairly very important for the complexification of the Funk-Hecke formulation, that is effectively used to introduce orthogonally invariant ideas of the lowered wave equation. The radial components of those suggestions are both Bessel or Hankel capabilities, which play an enormous function within the mathematical conception of acoustical and optical waves. those theories frequently require a close research of the asymptotic habit of the ideas. The offered creation of Bessel and Hankel services yields at once the major phrases of the asymptotics. Approximations of upper order will be deduced.

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The following proof is very direct and straightforward. It has seemingly not been used before. All proofs are based on operations in the neighborhood of the identity, or more precisely, by sequences of operators which tend to the identity. From now on the projections IPn(q, f) are always projections into the spaces Yn(q) and we call (IPn(q)f)(e) the Fourier-Laplace components of j, or the F-L projection of order n in We start with e. 24 1. The General Theory Definition 1: For f E C(Sq-l) we set with E( n) q, = (~) ~ 2 47r r(n+q-l) r(n +~) This operator approximates the identity, as the following theorem shows.

The numbers N(q, n) for the dimensions of the spaces y~(q) and the quantities Isq-11 for the areas of the unit spheres are indispensable in a theory of spherical symmetries in IRq. 9) ~ ~ 2 3 by Exercise 6,§3 q-2 N(q,n)-2- 1 (2n 2 +q- 2)(n + q - 3)! (q - 3)! (q; 2+n) (n +! 2) that the series in Lemma 2 satisfies for r E [0,1] q-2~ -2- L. 11) L rn cosncp = 00 n=l 1 +r 1_ r2 2 2 - rcoscp which is Lemma 2 with t = coscp and Pn(2jcoscp) = cosncp The Poisson identity is interesting because it provides a second proof of the completeness of the spherical harmonics.

1O) which again shows the homogeneous harmonics as residues. 12) Hn(x) = L IX(q)12kYn_2k(X) k=O with uniquely determined homogeneous harmonics Y n - 2k . As an application we now introduce the isotropically symmetric homogeneous harmonics as residues of (x· TJ)n, TJ E Sq-l. We have t::,k (x. 13) n! (x. TJ)n-2k (n - 2k)! 14) (xTJ)n - JD)n(q)(XTJ)n [~l = '"' ~ k=O (l)k __ 4 , r( n - k + ~) n. (n - 2k)! 2 Ix (q) 12k(x )n-2k TJ a harmonic that is invariant with regard to 3(q, TJ). 15) To determine en (q) we consider n even and n odd separately.