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By Claus Müller

This publication offers a brand new and direct method into the theories of precise capabilities with emphasis on round symmetry in Euclidean areas of ar­ bitrary dimensions. crucial elements will even be known as straightforward end result of the selected strategies. The primary subject is the presentation of round harmonics in a idea of invariants of the orthogonal team. H. Weyl used to be one of many first to show that round harmonics needs to be greater than a lucky wager to simplify numerical computations in mathematical physics. His opinion arose from his career with quan­ tum mechanics and used to be supported by way of many physicists. those rules are the major subject all through this treatise. whilst R. Richberg and that i all started this undertaking we have been shocked, how effortless and chic the final thought can be. one of many highlights of this ebook 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's effectively used to introduce orthogonally invariant suggestions of the diminished wave equation. The radial components of those options are both Bessel or Hankel features, which play a tremendous position within the mathematical idea of acoustical and optical waves. those theories usually require a close research of the asymptotic habit of the recommendations. The awarded advent of Bessel and Hankel capabilities yields without delay the top phrases of the asymptotics. Approximations of upper order could be deduced.

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R(~) . 7) (n+q-3)! t) + j + q - 3)! ). Lemma 2: Forq2":3,O:

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). With polar coordinates in the standard notation we get for the restriction to Sq-l. 6) 1;~21 Lq-2 (t + i~ e(q-l) . 71(q_l»)nlj(q - 1; 71(q_l»)dS,(;;)2 = P~ (q; t)lj(q - 1; e(q-l») This is an element of Yn(q). Lemma 1: Suppose lj(q - 1;·) E Yj(q - 1) then P~(q; t)lj(q - l;e(q-l») is a spherical harmonic of degree n in standard coordinates. 56 2.

Of §1O. We formalize this process by mappings of Ym(q - 1) into Yn(q) and introduce with the standard polar coordinates of Sq-1 the operator A;:". Definition 1: For q 2: 3 and m A;;:: Ym(q - 1) -> = 0,1, ... ))(~) := A:(q, t)Ym(q - 1; ~(q-1») The space Y;;'(q) : = A;:"(Ym(q - 1)) is called the associated space of order min Yn(q). We can now prove the main result of this section. 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.

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