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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.

42A10 Trigonometric approximation
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiń≠-type inequalities)
Full Text: DOI
[1] V. K. Dzyadyk, Introduction to the Theory of Uniform Approximation of Functions by Polynomials (Nauka, Moscow, 1977) [in Russian]. · Zbl 0481.41001
[2] M. G. Pleshakov, ”Comonotone Jackson’s inequality,” J. Approx. Theory 99(2), 409–421 (1999). · Zbl 0976.41016 · doi:10.1006/jath.1999.3327
[3] M. G. Pleshakov, Comonotone approximation of Periodic Functions of Sobolev Classes, Candidate’s Dissertation in Mathematics and Physics (Saratov State University, Saratov, 1997) [in Russian].
[4] A. S. Shvedov, ”Comonotone approximation of functions by polynomials,” Dokl. Akad. Nauk SSSR 250(1), 39–42 (1980) [Soviet Math. Dokl. 21, 34–37 (1980)]. · Zbl 0452.41004
[5] A. S. Shvedov, ”Orders of coapproximations of functions by algebraic polynomials,” Mat. Zametki 29(1), 117–130 (1981). · Zbl 0462.41006
[6] R. A. DeVore, D. Leviatan, and I. A Shevchuk, ”Approximation of monotone functions: a counter example,” in Curves and Surfaces with Application in CAGD, Chamonix-Mont-Blanc, 1996 (Vanderbilt Univ. Press, Nashville, TN, 1997), pp. 95–102. · Zbl 0958.41012
[7] G. G. Lorentz and K. L. Zeller, ”Degree of approximation by monotone polynomials, I,” J. Approx. Theory 1(4), 501–504 (1968). · Zbl 0172.07901 · doi:10.1016/0021-9045(68)90039-7
[8] A. S. Shvedov, ”Coapproximation of piecewise monotone functions by polynomials,” Mat. Zametki 30, 839–846 (1981). · Zbl 0485.41006
[9] H. Whitney, ”On functions with bounded nth differences,” J. Math. Pures Appl. (9) 36, 67–95 (1957). · Zbl 0077.06901
[10] P. A. Popov, ”An analog of the Jackson inequality for coconvex approximation of periodic functions,” Ukrain. Mat. Zh. 53(7), 919–928 (2001) [Ukrainian Math. J. 53 (7), 1093–1105 (2001)]. · Zbl 1002.41010
[11] P. A. Popov and M. G. Pleshakov, ”Sign-preserving approximation of periodic functions,” Ukrain. Mat. Zh. 55(8), 1087–1098 (2003) [Ukrainian Math. J. 55 (8), 1314–1328 (2003)]. · Zbl 1089.41006
[12] G. A. Dzyubenko, J. Gilewicz, and I. A. Shevchuk, ”Piecewise monotone pointwise approximation,” Constr. Approx. 14(3), 311–348 (1998). · Zbl 0912.41007 · doi:10.1007/s003659900077
[13] J. Gilewicz and I. A. Shevchuk, ”Comonotone approximation,” Fundam. Prikl. Mat. 2(2), 319–363 (1996). · Zbl 0908.41003
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