JACKSON TYPE THEOREMS FOR COMPACT RANK 1 SYMMETRIC SPACES

Abstract

Since recently various problems of approximation theory have been actively studied for functions on the n-dimensional sphere Sn (see [1–3] and the bibliography therein). A broader but natural class of spaces for which we can pose such problems is that of all compact rank 1 symmetric spaces (CROSP in the terminology of the monograph [4]). Some results are already available for these spaces (see [5–9]); however, the main problems still remain open. This article is devoted to proving the direct Jackson-type theorems for an arbitrary CROSP M . The complete classification of all CROSPs is well known. It comprises the four series: the spheres Sn (n = 1, 2, . . . ) and the real, complex, and quaternionic projective spaces (Pn(R), n = 2, 3, . . . ; Pn(C), n = 4, 6, . . . ; and Pn(H), n = 8, 12, 16, . . . ) (everywhere the superscript n denotes the dimension of the space) and one special space, the Cayley elliptic plane P 16(Cay). Since the problems of harmonic analysis on Pn(R) are easily reduced to the corresponding problems on Sn, we assume that M 6= Pn(R). A CROSP M is always a Riemannian manifold. Let ∆ be the Laplace–Beltrami operator on M . The spectrum of ∆ is discrete, real, and nonpositive. We arrange it in decreasing order (0 = λ0 > λ1 > λ2 > . . . ) and denote the eigensubspace of ∆ corresponding to the eigenvalue λk by Hk (it is always finite-dimensional). Put PN (M) := H0 + H1 + · · · + HN . The functions in PN (M) are referred to as spherical polynomials on M of degree m (for M = Sn they coincide with the ordinary spherical polynomials). Given a set X with a measure dσ, we as usual denote by Lp(X, dσ) the Banach space of complexvalued measurable functions f(x) on X with the finite norm