Abstract
We consider a 2π-periodic function f continuous on \(\mathbb{R}\) and changing its sign at 2s points y i ∈ [−π, π). For this function, we prove the existence of a trigonometric polynomial T n of degree ≤n that changes its sign at the same points y i and is such that the deviation | f(x) − T n(x) | satisfies the second Jackson inequality.
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REFERENCES
M. G. Pleshakov and P. A. Popov, “Sign-preserving approximation of periodic functions,” Ukr. Mat. Zh., 55, No. 8, 1087–1098 (2003).
I. A. Shevchuk, Polynomial Approximation and Traces of Functions Continuous on an Interval [in Russian], Naukova Dumka, Kiev (1992).
M. G. Pleshakov, Comonotone Approximation of Periodic Functions from Sobolev Classes [in Russian], Candidate-Degree Thesis (Physics and Mathematics), Saratov (1998).
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Pleshakov, M.G., Popov, P.A. Second Jackson Inequality in a Sign-Preserving Approximation of Periodic Functions. Ukrainian Mathematical Journal 56, 153–160 (2004). https://doi.org/10.1023/B:UKMA.0000031710.44467.5e
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DOI: https://doi.org/10.1023/B:UKMA.0000031710.44467.5e