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Comonotone approximation of periodic functions. (English) Zbl 1151.42001
Math. Notes 83, No. 2, 180-189 (2008); translation from Mat. Zametki 83, No. 2, 199-209 (2008).
Summary: Suppose that a continuous $$2\pi$$-periodic function $$f$$ on the real axis $$\mathbb R$$ changes its monotonicity at different ordered fixed points $$y_i \in [-\pi,\pi)$$, $$i = 1,\ldots,2s$$,$$s \in \mathbb N$$. In other words, there is a set $$Y:= \{y_i\}_{i\in \mathbb Z}$$ of points $$y_i = y_{i+2s} + 2\pi$$ on $$\mathbb R$$ such that, on $$[y_i,y_{i-1}]$$, $$f$$ is nondecreasing if $$i$$ is odd and nonincreasing if $$i$$ is even. For each $$n \geq N(Y)$$, we construct a trigonometric polynomial $$P_n$$ of order $$\leq n$$ changing its monotonicity at the same points $$y_i \in Y$$ as $$f$$ and such that $\left\| f-P_n \right\| \leqslant c(s)\omega_2 \left(f,\frac{\pi}{n}\right),$ where $$N(Y)$$ is a constant depending only on $$Y$$, $$c(s)$$ is a constant depending only on $$s$$, $$\omega_{2}(f,\cdot)$$ is the modulus of continuity of second order of the function $$f$$, and $$\|\cdot\|$$ is the max-norm.

MSC:
 42A10 Trigonometric approximation 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
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References:
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