Polynomial operators for spectral approximation of piecewise analytic functions

We construct a sequence of globally defined polynomial valued operators, using linear combinations of either the coefficients in the orthogonal polynomial expansion of the target function with respect to a very general mass distribution μ or the values of the function at suitable points on [−1, 1], so that the degree of approximation by these operators globally is commensurate with the degree of best polynomial approximation, and decays geometrically fast on intervals where the target function is analytic. Specifically, we construct linear operators σn, n = 0, 1, · · ·, on the space C[−1, 1] of continuous functions on [−1, 1] with the following properties: (1) For any f ∈ C[−1, 1], σn(f) is a polynomial of degree 8n. (2) If f is analytic on an interval I ⊆ [−1, 1], then maxx∈I |f(x)−σn(f, x)| ≤ c1 exp(−c2n) for positive constants c1, c2 which may depend upon f and I . (3) On the whole interval, maxx∈[−1,1] |f(x) − σn(f, x)| → 0 at a rate comparable with the distance of f from polynomials of degree at most n. We further address the question of detection of analytic singularities of f , based on non–adaptive data. Thus, we construct linear operators τn and degree 2 n+3 polynomials Ψk,n, k = 0, · · · , 2n+6, such that every f ∈ C[−1, 1] admits a uniformly convergent Littlewood–Paley type decomposition f = ∑∞ n=0 ∑ k τn(f, yk,n)Ψk,n for a suitable system of points yk,n, independent of f , with the following properties: (1) f is analytic at a point x0 ∈ [−1, 1] if and only if x0 is contained in a nondegenerate subinterval I ⊆ [−1, 1] such that lim supn→∞{maxyk,n∈I |τn(f, yk,n)|} n < 1. (2) A weighted `2 norm of the coefficients τn(f, yk,n) is of the same order of magnitude as the L(μ) norm of f . The results are applied to approximation on the sphere. ∗The research of this author was supported, in part, by grant DMS-0605209 from the National Science Foundation and grant W911NF-04-1-0339 from the U.S. Army Research Office. 1