By Isaac Chavel

The fundamental ambitions of the booklet are: (i) to introduce the topic to these attracted to researching it, (ii) to coherently current a couple of uncomplicated thoughts and effects, presently utilized in the topic, to these operating in it, and (iii) to give a few of the effects which are beautiful of their personal correct, and which lend themselves to a presentation now not overburdened with technical equipment.

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**Example text**

Given u l , . , U k - P = E L2(M),let inf ”fllllfl12 where f varies over the subspace (less the origin) of functions in 5 ( M ) orthogonal to ul, . Then, for eigenvalues given in (79), we have 1,. Of course, if u l , . , k - l , t h e n p = & . PROOF:Consider the functions f of the form k f 1 Uj4j9 j= 1 41,.. , are orthonormal, with each an eigenfunction of Aj, 1, . , k, and where f is orthogonal to ul, . , k - 1. j= 1 If we think of ul, . , ak as unknowns and (4j,ul) as given coefficients, then system (83) has more unknowns than equations and a nontrivial solution of (83) must exist.

It is known that all eigenfunctions of the sphere are obtained in this manner. More precisely, the space of homogeneous harmonic polynomials on Rn+' of degree k, when restricted to S",constitute the eigenspace of the kth distinct eigenvalue (14) X,, = k(k + n - l), where, now, k = 0,1,2,. , that is, we have labeled the eigenvalues as starting from 1, = 0. ] for details. We note for future reference. [l, p. 159 f]. and Stein-Weiss [l, p. PROPOSITION 1. , n + 11. u = -+ M, 0, then, by (2), @*f is also a solution, that is, a* preserves the eigenspaces of A.

Then, for eigenvalues given in (79), we have 1,. Of course, if u l , . , k - l , t h e n p = & . PROOF:Consider the functions f of the form k f 1 Uj4j9 j= 1 41,.. , are orthonormal, with each an eigenfunction of Aj, 1, . , k, and where f is orthogonal to ul, . , k - 1. j= 1 If we think of ul, . , ak as unknowns and (4j,ul) as given coefficients, then system (83) has more unknowns than equations and a nontrivial solution of (83) must exist. But then k p\lfl12 5 D[jlfl = 1 Saj’ 5 Akllfl12 j= 1 which implies the claim.