About Widths of Wiener Space in theLq-Norm

Abstract

We calculate the average Kolmogorov and linear n-widths of the Wiener space in the L q -norm. For the case 1 Յ q Ͻ ȍ, the n-widths d n decrease asymptotically as n Ϫ1/2 .

Approximation of SαS Lévy processes in Lp norm

“…We will only consider weak asymptotics: for two sequences a n , b n of real numbers, let us write a n b n iff lim n a n /b n < ∞, and a n b n iff a n b n a n . In [5,[12][13][14], the weak asymptotics for the case = 2 (i.e., the Wiener process) were determined. The results may be summarized as follows: Theorem 1.…”

Optimal approximation of stochastic differential equations by adaptive step-size control

“…Here c > 0 does only depend on σ and X(0). Moreover, we can apply Maiorov's result on average n-widths of the Wiener space, see Maiorov [7]. Here even complete knowledge of the trajectory of W is allowed.…”

Probabilistic and average linear widths of weighted Sobolev spaces on the ball equipped with a Gaussian measure

“…A few interesting results have been obtained. These include results on probabilistic and average Kolmogorov and linear widths of a Sobolev space of single-variate periodic functions with a Gaussian measure and the C r [0, 1] space equipped with the r-fold Wiener measure in the L q norm for 1 ≤ q ≤ ∞ (see [10,11,14,15,16,17,18,19,23,25]), of a space of multivariate periodic Sobolev functions equipped with a Gaussian measure in the L q norm for 1 < q < ∞ (see [3,4,28]), and of a Sobolev space W r 2 (M d−1 ) with a Gaussian measure on compact two-point homogeneous spaces M d−1 for 1 ≤ q ≤ ∞ (see [26]). In the worst and average case setting, the orders of the Kolmogorov and linear widths of weighted Sobolev classes on B d in L q,µ were presented in [27] and [29] respectively.…”