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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’skiń≠-type inequalities)
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