Comonotone approximation of twice differentiable periodic functions
Abstract
In the case where a 2π-periodic function f is twice continuously differentiable on the real axis ℝ and changes its monotonicity at different fixed points yi∈[−π,π),i=1,…,2s,s∈ℕ(i.e., on ℝ, there exists a set Y:=yii∈ℤ of points yi=yi+2s+2π such that the function f does not decrease on [yi,yi−1] if i is odd and does not increase if i is even), for any natural k and n,n≥N(Y,k)=const, we construct a trigonometric polynomial Tn of order ≤n that changes its monotonicity at the same points yi∈Y as f and is such that ∥f−Tn∥≤c(k,s)n2ωk(f″,1/n) (∥f−Tn∥≤c(r+k,s)nrωk(f(r),1/n),f∈C(r),r≥2), where N(Y,k) depends only on Y and k,c(k,s) is a constant depending only on k and s,ωk(f,⋅) is the modulus of smoothness of order k for the function f, and ‖⋅‖ is the max-norm.Downloads
Published
25.04.2009
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Section
Research articles
How to Cite
Dzyubenko, H. A. “Comonotone Approximation of Twice Differentiable Periodic Functions”. Ukrains’kyi Matematychnyi Zhurnal, vol. 61, no. 4, Apr. 2009, pp. 435-51, https://umj.imath.kiev.ua/index.php/umj/article/view/3032.