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

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Abstract

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.

Original languageEnglish
Pages (from-to)303-317
Number of pages15
JournalJournal of Mathematical Sciences (United States)
Volume234
Issue number3
DOIs
StatePublished - Oct 2018

Scopus subject areas

  • Statistics and Probability
  • Mathematics(all)
  • Applied Mathematics

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