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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’skiń≠-type inequalities)
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