TY - JOUR

T1 - A Sharp Jackson–Chernykh Type Inequality for Spline Approximations on the Line

AU - Vinogradov, O. L.

PY - 2020/1/1

Y1 - 2020/1/1

N2 - An analog of the Jackson–Chernykh inequality for spline approximations in the space L2( ℝ) is established in this work. For r ℕ and 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 of minimal defect with knots , j ∏/σ , and by ω( f, δ)2 the first-order modulus of continuity of f in L2( ℝ). The main result of our work is as follows. For any f ∈ L2( ℝ),(formula presented) where θr = 1/√1-εr 2 and εr is the positive root of the equation(formula presented) The constant 1/√2 cannot be reduced on the whole class L2(ℝ ) even by increasing the step of the modulus of continuity.

AB - An analog of the Jackson–Chernykh inequality for spline approximations in the space L2( ℝ) is established in this work. For r ℕ and 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 of minimal defect with knots , j ∏/σ , and by ω( f, δ)2 the first-order modulus of continuity of f in L2( ℝ). The main result of our work is as follows. For any f ∈ L2( ℝ),(formula presented) where θr = 1/√1-εr 2 and εr is the positive root of the equation(formula presented) The constant 1/√2 cannot be reduced on the whole class L2(ℝ ) even by increasing the step of the modulus of continuity.

KW - Jackson inequality

KW - sharp constants

KW - splines

UR - http://www.scopus.com/inward/record.url?scp=85082629901&partnerID=8YFLogxK

U2 - 10.1134/S1063454120010112

DO - 10.1134/S1063454120010112

M3 - Article

AN - SCOPUS:85082629901

VL - 53

SP - 10

EP - 19

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 1

ER -