Geodesic
Pointwise approximation of periodic functions by trigonometric polynomials with Hermitian interpolation
Izvestiya. Mathematics , Tome 73 (2009) no. 4, pp. 699-726.

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We prove a general direct theorem on the simultaneous pointwise approximation of smooth periodic functions and their derivatives by trigonometric polynomials and their derivatives with Hermitian interpolation. We study the order of approximation by polynomials whose graphs lie above or below the graph of the function on certain intervals. We prove several inequalities for Hermitian interpolation with absolute constants (for any system of nodes). For the first time we get a theorem on the best-order approximation of functions by polynomials with interpolation at a given system of nodes. We also provide a construction of Hermitian interpolating trigonometric polynomials for periodic functions (in the case of one node, these are trigonometric Taylor polynomials).
Keywords: trigonometric Taylor polynomial, best approximation, modulus of smoothness, two-sided approximation estimates, piecewise one-sided approximation, factorization of differential operators. 
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R. M. Trigub. Pointwise approximation of periodic functions by trigonometric polynomials with Hermitian interpolation. Izvestiya. Mathematics , Tome 73 (2009) no. 4, pp. 699-726. http://geodesic.mathdoc.fr/item/IM2_2009_73_4_a3/

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