Operators and Harmonic Analysis on the Sphere
1966; American Mathematical Society; Volume: 125; Issue: 2 Linguagem: Inglês
10.2307/1994352
ISSN1088-6850
Autores Tópico(s)Algebraic and Geometric Analysis
ResumoIntroduction.The main result of this paper concerns operators which commute with all rotations on certain spaces of functions on Sk, the /¿-dimensional sphere (k^2), namely C, L1, L".The proofs use harmonic analysis of various spaces of functions and measures on Sk, which involves the ultraspherical polynomials.Notation.Sk admits a group of rotations, namely the special orthogonal group SOfc + 1.The result of the action of the rotation a on the point x will be denoted by xa.The "rotation" operator Ra acting on functions (f) and measures (/*) is defined by RJ(x) = /(*«) for all x e Sk, Rap(E) = n(Ea) for all /¿-measurable subsets E of Sk.P% is the ultraspherical polynomial of index A and degree n (normalized by ^nO)=l)-Considering Sk imbedded as the unit sphere in Rk + 1, let x-y be the ordinary inner product of the vectors which correspond to the points x and y (-1 ^x-y¿ 1).Sk has a unique rotation invariant Borel measure, say m, such that m(Sk)=l, and the use of this measure is implied by notations such as "dx"; further let Lp(Sk)=Lp (Sk, ni).Statement of results.With each ¡j, e M(Sk), the space of finite regular Borel measures on Sk, and each /eL1^'), the following continuous functions are associated : fin(x) = jgk Pf -»'2(x y)My) " = 0, 1, 2,... and /"(*) = jsh n-iV\x-y)f{y) dy n = 0, 1, 2,... respectively.The sequence {fin} [{/"}] determines /*[/] uniquely.We choose a north
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