Optimal information for approximating periodic analytic functions

Let Sβ := {z ∈ C : | Im z| < β} be a strip in the complex plane. For fixed integer r ≥ 0 let Hr ∞,β denote the class of 2π-periodic functions f , which are analytic in Sβ and satisfy |f(r)(z)| ≤ 1 in Sβ . Denote by Hr,R ∞,β the subset of functions from Hr ∞,β that are real-valued on the real axis. Given a function f ∈ Hr ∞,β , we try to recover f(ζ) at a fixed point ζ ∈ R by an algorithm A on the basis of the information If = (a0(f), a1(f), . . . , an−1(f), b1(f), . . . , bn−1(f)), where aj(f), bj(f) are the Fourier coefficients of f . We find the intrinsic error of recovery E(H ∞,β , I) := inf A : C2n−1→C sup f∈Hr ∞,β |f(ζ)− A(If)|. Furthermore the (2n−1)-dimensional optimal information error, optimal sampling error and n-widths of Hr,R ∞,β in C, the space of continuous functions on [0, 2π], are determined. The optimal sampling error turns out to be strictly greater than the optimal information error. Finally the same problems are investigated for the class Hp,β , consisting of all 2π-periodic functions, which are analytic in Sβ with p-integrable boundary values. In the case p = 2 sampling fails to yield optimal information as well in odd as in even dimensions. Introduction Let W be a class of 2π-periodic, real-valued (or complex-valued) functions. Suppose that W ⊂ C, where C is the space of continuous functions on [0, 2π] endowed with the supremum norm. Consider the problem of optimal recovery of the linear functional U on W given by Uf = f(ζ), i.e. point evaluation in ζ, on the basis of the information If = (L1f, . . . , Lnf), where L1, . . . , Ln are continuous linear functionals on W . By an algorithm we mean any map (not necessarily linear or continuous) A : Z → Z, where Z = R or C depending on whether W is a set of real-valued or complex-valued functions. Received by the editor March 25, 1996. 1991 Mathematics Subject Classification. Primary 65E05, 41A46; Secondary 30E10.