Точное неравенство типа Джексона - Черныха для приближений сплайнами на оси.

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Abstract

An analog of the Jackson Chernykh inequality for spline approximations in the space L2(ℝ) is established. For r ∈ ℕ, σ > 0, we denote by Aστr(f)2 the best approximation of a function f ∈ L2(ℝ) by the space of splines of degree r and of minimal defect with knots jπ/σ, j ∈ ℤ, and by ω(f, δ)2 its first order modulus of continuity in L2(ℝ). The main result of the paper is the following. For every f ∈ L2(ℝ) Aσr(f)2 ≤ 1/√2 ω (f, θrπ/σ)2, where εr is the positive root of the equation (4ε2(ch πε/τ-1))/(ch πε/τ+cos π/τ)-1/(32r-2), τ = √(1-ε2), θr = 1/√(1-ε 2/r). The constant 1/√2 cannot be reduced on the whole class L2(ℝ), even if one insreases the step of the modulus of continuity.
Original languageRussian
Pages (from-to)15-27
JournalВЕСТНИК САНКТ-ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА. МАТЕМАТИКА. МЕХАНИКА. АСТРОНОМИЯ
Volume7
Issue number1
StatePublished - 2020
Externally publishedYes

Keywords

  • Jackson inequality
  • sharp constants
  • splines
  • неравенство Джексона
  • сплайны
  • точные константы

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