TY - JOUR

T1 - Sharp constants for approximations of convolution classes with an integrable kernel by spaces of shifts

AU - Vinogradov, O. L.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - Let σ > 0, and let G,B ∈ L(ℝ). This paper is devoted to approximation of convolution classes f = ϕ*G, ϕ ∈ Lp(ℝ), by a space SB that consists of functions of the form. Under some conditions on G and B, linear operators Xσ,G,B with values in SB are constructed for which ||f -Xσ,G,B(f)||p ≤ Kσ,G||ϕ||p. For p = 1,∞ the constant Kσ,G (it is an analog of the well-known Favard constant) cannot be reduced, even if one replaces the left-hand side by the best approximation by the space SB. The results of the paper generalize classical inequalities for approximations by entire functions of exponential type and by splines.

AB - Let σ > 0, and let G,B ∈ L(ℝ). This paper is devoted to approximation of convolution classes f = ϕ*G, ϕ ∈ Lp(ℝ), by a space SB that consists of functions of the form. Under some conditions on G and B, linear operators Xσ,G,B with values in SB are constructed for which ||f -Xσ,G,B(f)||p ≤ Kσ,G||ϕ||p. For p = 1,∞ the constant Kσ,G (it is an analog of the well-known Favard constant) cannot be reduced, even if one replaces the left-hand side by the best approximation by the space SB. The results of the paper generalize classical inequalities for approximations by entire functions of exponential type and by splines.

KW - Akhiezer-Kreĭn-Favard type inequalities

KW - Convolution

KW - Sharp constants

KW - Spaces of shifts

UR - http://www.scopus.com/inward/record.url?scp=85070547867&partnerID=8YFLogxK

U2 - 10.1090/spmj/1572

DO - 10.1090/spmj/1572

M3 - Article

AN - SCOPUS:85070547867

VL - 30

SP - 841

EP - 867

JO - St. Petersburg Mathematical Journal

JF - St. Petersburg Mathematical Journal

SN - 1061-0022

IS - 5

ER -