Sharp Jackson — Chernykh type inequality for spline approximations on the line

Abstract

An analog of the Jackson — Chernykh inequality for spline approximations in the space L2(R) is established. For r ∈ N, σ > 0, we denote by Aσr(f)2 the best approximation of a function f ∈ L2(R) by the space of splines of degree r and of minimal defect with knots jπ σ , j ∈ Z, and by ω(f, δ) its first order modulus of continuity in L2(R). The main result of the paper is the following. For every f ∈ L2(R)
Aσr(f)2 6 1 √ 2 ω f, θrπ σ 2 ,
where εr is the positive root of the equation
4ε  2 (ch πε τ − 1) ch πε τ + cos π τ = 1 3 2r−2 , τ = p 1 − ε 2, θr = √ 1 1−ε2 r.

The constant √12 cannot be reduced on the whole class L2(R), even if one insreases the step of the modulus of continuity.

Keywords:

Jackson inequality, splines, sharp constants