Abstract
We prove the Jackson theorem for a zero-preserving approximation of periodic functions (i.e., in the case where the approximating polynomial has the same zeros y i) and for a sign-preserving approximation [i.e., in the case where the approximating polynomial is of the same sign as a function f on each interval (y i, y i − 1)]. Here, y i are the points obtained from the initial points −π ≤ y 2s ≤y 2s−1 <...< y1 < π using the equality yi = yi + 2s + 2π furthermore, these points are zeros of a 2π-periodic continuous function f.
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Pleshakov, M.G., Popov, P.A. Sign-Preserving Approximation of Periodic Functions. Ukrainian Mathematical Journal 55, 1314–1328 (2003). https://doi.org/10.1023/B:UKMA.0000010761.91730.16
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DOI: https://doi.org/10.1023/B:UKMA.0000010761.91730.16