One-sided widths of classes of smooth functions

Yurii N Subbotin

doi:10.15826/umj.2015.1.008

Abstract

One-sided widths of the classes of functions W p r [0,1] in the metric L q [0,1], 1≤ p, q ≤ ∞, r ≥ 1 are studied. Such widths are defined similarly to Kolmogorov widths with additional constraints on the approximating functions.

Highlights

Ln⊂Lq sup f ∈Wpr inf g(x)∈Ln f −g where Ln is an n-dimensional subspace of the space Lq[0, 1]; Wpr is the class of functions f (x) representable in the form x f (x)
We show that one-sided widths d+n (Wpr, Lq) have the same orders (2) with respect to n
D+[ mr ]r+1(Wpr, Lq) ≤ d+m(Wpr, Lq) ≤ d+[ mr ]r(Wpr, Lq) and, from the foregoing, we obtain the exact order of behavior of the one-sided widths with respect to m (m → ∞) for all m, for m that are multiples of r

Summary

Introduction

Where Ln is an n-dimensional subspace of the space Lq[0, 1]; Wpr is the class of functions f (x) representable in the form x f (x). The Kolmogorov width (see [1]) is, by definition, the value dn (Wpr , Lq ) Ln⊂Lq sup f ∈Wpr inf g(x)∈Ln f −g where Ln is an n-dimensional subspace of the space Lq[0, 1]; Wpr is the class of functions f (x) representable in the form x f (x) The corresponding one-sided width is defined as follows (see [2]): d+n (Wpr, Lq)

Results

Conclusion