MULTIVARIATE INTEGRATION OF INFINITELY MANY TIMES DIFFERENTIABLE FUNCTIONS IN WEIGHTED KOROBOV SPACES

作者：Kritzer Peter^*; Pillichshammer Friedrich; Wozniakowski Henryk

来源：Mathematics of Computation, 2014, 83(287): 1189-1206.

We study multivariate integration for a weighted Korobov space of periodic infinitely many times differentiable functions for which the Fourier coefficients decay exponentially fast. The weights are defined in terms of two non-decreasing sequences a = {a(i)} and b = {b(i)} of numbers no less than one and a parameter omega is an element of(0, 1). Let e(n, s) be the minimal worst-case error of all algorithms that use n function values in the s-variate case. We would like to check conditions on a, b and. such that e(n, s) decays exponentially fast, i.e., for some q is an element of(0,1) and p %26gt; 0 we have e(n, s) = O(q(np)) as n goes to infinity. The factor in the O notation may depend on s in an arbitrary way. We prove that exponential convergence holds iff B :Sigma(infinity)(i=1) 1/b(i) %26lt; infinity independently of a and omega. Furthermore, the largest p of exponential convergence is 1/B. We also study exponential convergence with weak, polynomial and strong polynomial tractability. This means that e(n, s) = C(s) q(np) for all n and s and with log C(s) = exp(o(s)) for weak tractability, with a polynomial bound on log C(s) for polynomial tractability, and with uniformly bounded C(s) for strong polynomial tractability. We prove that the notions of weak, polynomial and strong polynomial tractability are equivalent, and hold iff B %26lt; infinity and a(i) are exponentially growing with i. We also prove that the largest (or the supremum of) p for exponential convergence with strong polynomial tractability belongs to [1/(2B), 1/B].

• 出版日期2014-5