# zbMATH — the first resource for mathematics

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)
Full Text:
##### References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.