Analysis of Spherical Symmetries in Euclidean Spaces by Claus Müller

By Claus Müller

This booklet offers a brand new and direct process into the theories of designated features with emphasis on round symmetry in Euclidean areas of ar­ bitrary dimensions. crucial components will even be known as straight forward end result of the selected options. The critical subject is the presentation of round harmonics in a thought of invariants of the orthogonal staff. H. Weyl used to be one of many first to show that round harmonics has to be greater than a lucky bet to simplify numerical computations in mathematical physics. His opinion arose from his career with quan­ tum mechanics and was once supported by way of many physicists. those principles are the major topic all through this treatise. whilst R. Richberg and that i began this venture we have been stunned, how effortless and chic the overall thought should be. one of many highlights of this publication is the extension of the classical result of round harmonics into the complicated. this can be really very important for the complexification of the Funk-Hecke formulation, that is effectively used to introduce orthogonally invariant suggestions of the decreased wave equation. The radial elements of those strategies are both Bessel or Hankel capabilities, which play an enormous position within the mathematical thought of acoustical and optical waves. those theories usually require a close research of the asymptotic habit of the options. The offered advent of Bessel and Hankel capabilities yields without delay the major phrases of the asymptotics. Approximations of upper order could be deduced.

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P ( . t) 2q-2 n ! n q, §1l The Associated Spaces y~(q) 55 §11 The Associated Spaces y~ (q) There are two aspects of the subject of this section. We may see it as a process to generate orthonormalized bases in a rational way, but we may also regard it in view of symmetry properties, related to the isotropical group J(q, eq). We follow the historical way and generalize Laplace's first integral. 1) a(q) = eq . 2) homogeneous harmonics, depending on 71(q-l). 3) is a homogenous and harmonic polynomial of degree n, and consequently an element of y~(q).

Theorem 1: Each space Yn(q), q 2: 3, n 2: 0 is the orthogonal direct sum of the associated spaces Y;;'(q) Proof: First, we show that the associated spaces in Yn(q) are mutually orthogonal. For 0 :0:::; k,m :0:::; n, Yk(q - l;'),Ym (q -1;·) and the scalar products < , >(q) ; < , >(q-1) we have (§11. ) = < Yk, Ym >(q-1) 1 >(q) A~(q; t) A;;:(q;t)(l- t2)~dt Thus Y! 1 Y;;' for k -=I- m. Moreover A~ is a bijection of Yk (q - 1) onto y~(q). 9) u=O N(q,n)u n l+u (1 - U)q-1 l+u 1 (1 - U)q-2 1 - u §11 The Associated Spaces y~ (q) 57 Since we are dealing with two linear spaces of equal dimension N(q, n), the assertion is proved.

Let us assume we have an orthonormal basis Y n ,l, Y n ,2, ... , Yn,N of Yn(q). We can find a system 0:1, 0:2, ... 2) Y n ,2(o:d ;k = 1, ... ,N-l 32 1. The General Theory can be constructed recursively. 4) Yn(~) = L CjYn,j(~) j=l and the coefficients are uniquely determined if we presecibe the values in at, ... 3) without knowing the elements Yn,j. This leads to Definition 1: A system of N(q, n) points al,"" aN on Sq-1 is called regular of degree n if the N x N determinant is positive. The system is called singular if the determinant is zero.

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