# zbMATH — the first resource for mathematics

Comonotone approximation of twice differentiable periodic functions. (Ukrainian, English) Zbl 1224.42003
Ukr. Mat. Zh. 61, No. 4, 435-451 (2009); translation in Ukr. Math. J. 61, No. 4, 519-540 (2009).
Summary: In the case where a $$2\pi$$-periodic function $$f$$ is twice continuously differentiable on the real axis $$\mathbb R$$ and changes its monotonicity at different fixed points $$y_i\in[-\pi,\pi), i=1,\dots,2s, s\in\mathbb N$$ (i.e., on $$\mathbb R$$ there exists a set $$Y:=\{y_i \}_{i\in\mathbb Z}$$ of points $$y_i=y_{i+2s}+2\pi$$ such that the function $$f$$ does not decrease on $$[y_ i,y_{i-1}]$$ if $$i$$ is odd and does not increase if $$i$$ is even), for any natural $$k$$ and $$n, n\geq N(Y, k) =\text{const}$$, we construct a trigonometric polynomial $$T_n$$ of order $$\leq n$$ that changes its monotonicity at the same points $$y_i\in Y$$ as $$f$$ and is such that $\| f-T_n\| \leq\frac{c(k,s)}{n^2}\omega_k(f'',1/n)$ $\left(\| f-T_n\| \leq\frac{c(r+k,s)}{n^r}\omega_k(f^{(r)},1/n),f\in C^{(r)},r\geq2\right),$ where $$N(Y, k)$$ depends only on $$Y$$ and $$k$$, $$c(k, s)$$ is a constant depending only on $$k$$ and $$s$$, $$\omega_k(f,\cdot)$$ is the modulus of smoothness of order $$k$$ for the function $$f$$, and $$\| \cdot\|$$ is the max-norm.

##### MSC:
 42A10 Trigonometric approximation 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
##### Keywords:
approximation; periodic function
Full Text: