TY - JOUR

T1 - Sharp Estimates of Linear Approximations by Nonperiodic Splines in Terms of Linear Combinations of Moduli of Continuity

AU - Vinogradov, O. L.

AU - Gladkaya, A. V.

N1 - Vinogradov, O.L., Gladkaya, A.V. Sharp Estimates of Linear Approximations by Nonperiodic Splines in Terms of Linear Combinations of Moduli of Continuity. J Math Sci 234, 303–317 (2018). https://doi.org/10.1007/s10958-018-4006-7

PY - 2018/10

Y1 - 2018/10

N2 - Assume that σ > 0, r, μ 휖 ℕ, μ ≥ r + 1, r is odd, p 휖 [1,+∞], and f∈Wp(r)(ℝ). We construct linear operators Xσ,r,μ whose values are splines of degree μ and of minimal defect with knots kπσ,k∈ℤ, such that ‖f−Xσ,r,u(f)‖p≤(πσ)r{Ar,02ω1|(f(r)πσ)p+∑v=1u−r−1Ar,vωv(f(r)πσ)p}+(πσ)r(Krπr−∑v=0u−r−12vAr,v)2r−μωμ−r(f(r)πσ)p, where for p = 1,.. ,+∞, the constants cannot be reduced on the class Wp(r)(ℝ). Here Kr=4π∑l=0∞(−1)l(r+1)(2l+1)r+1 are the Favard constants, the constants Ar,ν are constructed explicitly, and ωv is a modulus of continuity of order ν. As a corollary, we get the sharp Jackson type inequality‖f−Xσ,r,μ(f)‖p≤Kr2σrω1(f(r)πσ)p.

AB - Assume that σ > 0, r, μ 휖 ℕ, μ ≥ r + 1, r is odd, p 휖 [1,+∞], and f∈Wp(r)(ℝ). We construct linear operators Xσ,r,μ whose values are splines of degree μ and of minimal defect with knots kπσ,k∈ℤ, such that ‖f−Xσ,r,u(f)‖p≤(πσ)r{Ar,02ω1|(f(r)πσ)p+∑v=1u−r−1Ar,vωv(f(r)πσ)p}+(πσ)r(Krπr−∑v=0u−r−12vAr,v)2r−μωμ−r(f(r)πσ)p, where for p = 1,.. ,+∞, the constants cannot be reduced on the class Wp(r)(ℝ). Here Kr=4π∑l=0∞(−1)l(r+1)(2l+1)r+1 are the Favard constants, the constants Ar,ν are constructed explicitly, and ωv is a modulus of continuity of order ν. As a corollary, we get the sharp Jackson type inequality‖f−Xσ,r,μ(f)‖p≤Kr2σrω1(f(r)πσ)p.

UR - http://www.scopus.com/inward/record.url?scp=85052727426&partnerID=8YFLogxK

U2 - 10.1007/s10958-018-4006-7

DO - 10.1007/s10958-018-4006-7

M3 - Article

AN - SCOPUS:85052727426

VL - 234

SP - 303

EP - 317

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 3

ER -