Comonotone approximation of twice differentiable periodic functions

Authors

  • H. A. Dzyubenko

Abstract

In the case where a 2π-periodic function f is twice continuously differentiable on the real axis and changes its monotonicity at different fixed points yi[π,π),i=1,,2s,s(i.e., on , there exists a set Y:=yii of points yi=yi+2s+2π such that the function f does not decrease on [yi,yi1] if i is odd and does not increase if i is even), for any natural k and n,nN(Y,k)=const, we construct a trigonometric polynomial Tn of order n that changes its monotonicity at the same points yiY as f and is such that fTn∥≤c(k,s)n2ωk(f,1/n) (fTn∥≤c(r+k,s)nrωk(f(r),1/n),fC(r),r2), 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.

Published

25.04.2009

Issue

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.