Almost coconvex approximation of continuous periodic functions
Abstract
If a 2π -periodic function f continuous on the real axis changes its convexity at 2s,s∈N, points yi:π≤y2s<y2s−1<...<y1<π , and, for all other i∈Z, yi are periodically defined, then, for every natural n \geq N_{y_i}}, we determine a trigonometric polynomial Pn of order cn such that Pn has the same convexity as f everywhere except, possibly, small neighborhoods of the points yi:(yi\pi/n,yi+π/n), and ‖fPn‖≤c(s)ω4(f,π/n),, where N_{y_i}} is a constant depending only on mini=1,...,2s{yiyi+1},c and c(s) are constants depending only on s,ω4(f,⋅) is the fourth modulus of smoothness of the function f, and ‖⋅‖ is the max-norm.
Published
25.03.2019
How to Cite
Dzyubenko, H. A. “Almost Coconvex Approximation of Continuous Periodic Functions”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, no. 3, Mar. 2019, pp. 353-67, https://umj.imath.kiev.ua/index.php/umj/article/view/1444.
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Section
Research articles