By Claus Müller

ISBN-10: 1461205816

ISBN-13: 9781461205814

ISBN-10: 1461268273

ISBN-13: 9781461268277

This publication offers a brand new and direct strategy into the theories of particular features with emphasis on round symmetry in Euclidean areas of ar­ bitrary dimensions. crucial components will also be known as simple as a result selected innovations. The relevant subject is the presentation of round harmonics in a conception 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 profession with quan­ tum mechanics and was once supported via many physicists. those principles are the top topic all through this treatise. while R. Richberg and that i begun this venture we have been stunned, how effortless and stylish the final conception will be. one of many highlights of this booklet is the extension of the classical result of round harmonics into the advanced. this is often fairly very important for the complexification of the Funk-Hecke formulation, that is effectively used to introduce orthogonally invariant strategies of the lowered wave equation. The radial components of those strategies are both Bessel or Hankel features, which play an incredible function within the mathematical thought of acoustical and optical waves. those theories usually require an in depth research of the asymptotic habit of the strategies. The offered advent of Bessel and Hankel features yields at once the prime phrases of the asymptotics. Approximations of upper order may be deduced.

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Example text

Thus Ln is the Legendre harmonic of degree n and dimension q. From Lemma 2, §7 we get where ~(q-l) is any point of Sq-2 and t E [-1,1]. Evaluation ofthe integral in standard coordinates now yields Laplace's first integral. 42 2. The Specific Theories Theorem 1 will be referred to as the Laplace representation of the Legendre polynomials. To demonstrate its usefulness we derive another recurrence formula for the Pn(q; t) as well as some estimates. 8) Next, we prove inequalities for Pn(q; t). 8) this estimate remains valid for q = 2.

5) is zero and we have a nontrivial set of constants Ct, C 2, . 4) vanishes in all §6 Homogeneous Harmonics 33 al, ... ,aN. Therefore a system is singular if there is an element of Yn(q) that is zero in all N(q, n) points. Conversely, we have Lemma 1: Suppose al, ... , aN is a regular system of degree n; then every Y n E Yn(q), with Yn(aj) = O,j = 1, ... , N, vanishes identically. Of course not every system of N points is regular, although singularity is somehow the exception. The functions Pn(q; t; ·aj),j = 1, ...

The Specific Theories because the sum over j yields N~~~~;I) Pm{q - 1; ~(q-l) addition theorem in (q - 1) dimensions. We set u : = ~(q-l) . 13) to Lemma 2: For t, s, u E [-1,1] and q to by the 3 we have ~ ~) N(q,n)p (. Isq-11 ~ . T'/(q-l») nq,st+uVl-s~vl-t- = ISql_21 N(q - 1, m)A~(q, t)A~(q, s)Pn(q - 1; u) We multiply by Pdq -1; u)(l- U2)~ on both sides and integrate over [-1,1]. 17) We have now gained a rather closed and explicit theory of complete systems of functions on spheres, and it is an interesting question if this knowledge can be used in other situations.

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