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 2 N, σ > 0, we denote by A r(f)2 the best approximation of a function f 2 L2(R) by the space of splines of degree r and of minimal defect with knots j , j 2 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 2 L2(R) A r(f)2 6 1 p2 ω f, θrπ σ 2 , where εr is the positive root of the equation 4ε2(ch " − 1) ch " + cos

= 1 32r−2 , τ = p1 − ε2, θr = 1 p1−"2 r . The constant 1 p2 cannot be reduced on the whole class L2(R), even if one insreases the step of the modulus of continuity.