Average Widths of Sobolev Classes on Rn

Widths and shape-preserving widths of Sobolev-type classes of s-monotone functions

Abstract. Let I be a finite interval, r, n ∈ N, s ∈ N0 and 1 ≤ p ≤ ∞. Given a set M , of functions defined on I, denote by ∆+M the subset of all functions y ∈ M such that the s-difference ∆τ y(·) is nonnegative on I, ∀τ > 0. Further, denote by W r p the Sobolev class of functions x on I with the seminorm ‖x‖Lp ≤ 1. We obtain the exact orders of the Kolmogorov and the linear widths, and of the s...

SHAPE PRESERVING WIDTHS OF SOBOLEV-TYPE CLASSES OF s-MONOTONE FUNCTIONS ON A FINITE INTERVAL

Let I be a finite interval and r ∈ N. Denote by ∆ s + L q the subset of all functions y ∈ L q such that the s-difference ∆ s τ y(·) is nonnegative on I, ∀τ > 0. Further, denote by ∆ s + W r p , the class of functions x on I with the seminorm x (r) L p ≤ 1, such that ∆ s τ x ≥ 0, τ > 0. For s = 3,. .. , r + 1, we obtain two-sided estimates of the shape preserving widths

Comparison theorems of Kolmogorov type and exact values of n-widths on Hardy-Sobolev classes

Let Sβ := {z ∈ C : |Imz| < β} be a strip in complex plane. H̃r ∞,β denotes those 2π-periodic, real-valued functions on R which are analytic in the strip Sβ and satisfy the condition |f (r)(z)| ≤ 1, z ∈ Sβ . Osipenko and Wilderotter obtained the exact values of the Kolmogorov, linear, Gel′fand, and information n-widths of H̃r ∞,β in L∞[0, 2π], r = 0, 1, 2, . . ., and 2n-widths of H̃r ∞,β in Lq [0, ...

Kolmogorov and Linear Widths of Weighted Sobolev-Type Classes on a Finite Interval, II

Probabilistic and average linear widths of Sobolev space with Gaussian measure

E-mail: [email protected] Abstract: We determine the exact order of the p-average linear n-widths λ n (W r 2 , μ, Lq)p, 1 ≤ q < ∞, 0 < p <∞, of the Sobolev space W r 2 equipped with a Gaussian measure μ in the Lq-norm. Moreover, we also calculate the probabilistic linear (n, δ)-widths and p-average linear n-widths of the finitedimensional space R with the standard Gaussian measure in l q , i.e.,