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On the connection between the best approximation by algebraic polynomials and the modulus of smoothness of order \(r\). (English. Russian original) Zbl 1178.41014
J. Math. Sci., New York 155, No. 1, 153-169 (2008); translation from Sovrem. Mat., Fundam. Napravl. 25, 149-164 (2007).
For \(2\pi\)-periodic functions, there is a well-known relation between the \(r\)th modulus of smoothness \(\omega_r(f,\delta)_p\) of a function in \(L_p([0,2\pi])\) and its best approximations \(E_n(f)_p\) by trigonometric polynomials of degree less than or equal to \(n-1\).
Considering nonperiodic functions on a finite subinterval of the real axis, one fails to establish such relations between usual moduli of smoothness of these functions and their best approximations by algebraic polynomials. However, using generalized moduli of smoothness that are defined by a family of asymmetric generalized translation operators, an analogous relation to best approximations by algebraic polynomials can be shown.
MSC:
41A30 Approximation by other special function classes
41A50 Best approximation, Chebyshev systems
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