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

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 Weighted Sobolev-type Classes of Positive, Monotone and Convex Functions on a Finite Interval

Let X be a real linear space of vectors x with a norm ‖x‖X , W ⊂ X, W 6= ∅ and V ⊂ X, V 6= ∅. Let L be a subspace in X of dimension dim L ≤ n, n ≥ 0 and M = M(z) := z + L be a shift of the subspace L by an arbitrary vector z ∈ X. If M ∩ V 6= ∅, then we denote by E(x, M ∩ V )X := inf y∈Mn∩V ‖x− y‖X , the best approximation of the vector x ∈ X by M ∩ V , and by E(W,M ∩ V )X := sup x∈W E(x,M ∩ V )X ,

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

The degree of shape preserving weighted polynomial approximation

We analyze the degree of shape preserving weighted polynomial approximation for exponential weights on the whole real line. In particular, we establish a Jackson type estimate. Keywords: Shape Preserving Polynomials, k-Monotone, Exponential Weights, Jackson Theorem, Freud Weights. AMS Classi…cation: 41A29, 41A17 Research supported by NSF grant DMS1001182 and US-Israel BSF grant 2008399 1. Intro...

Optimal spline spaces of higher degree for L2 n-widths

In this paper we derive optimal subspaces for Kolmogorov n-widths in the L2 norm with respect to sets of functions defined by kernels. This enables us to prove the existence of optimal spline subspaces of arbitrarily high degree for certain classes of functions in Sobolev spaces of importance in finite element methods. We construct these spline spaces explicitly in special cases. Math Subject C...