## 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
Full Text:

### References:

 [1] H. Bateman and A. Erdelyi, Higher Transcendental Functions, Vol. 2 [Russian translation], Nauka, Moscow (1966). · Zbl 0143.29202 [2] P. L. Butzer, R. L. Stens, and M. Wehrens, ”Higher order of continuity based on the Jacobi translation operator and best approximation,” C. R. Math. Rep. Acad. Sci. Canada, 2, 83–87 (1980). · Zbl 0436.41013 [3] S. Pawelke, ”Ein Satz von Jacksonschen Typ für algebraische Polynome,” Acta Sci. Math., 33, No. 3–4, 323–336 (1972). · Zbl 0243.41005 [4] M. K. Potapov, ”Some inequalities for polynomials and their derivatives,” Vestn. MGU. Ser. Mat., No. 2, 10–20 (1960). [5] M. K. Potapov, ”On approximation by algebraic polynomials in the integral metric with Jacobian weight,” Vestn. MGU. Ser. Mat., No. 4, 43–52 (1983). · Zbl 0527.41003 [6] M. K. Potapov and F. M. Berisha, ”On Jackson’s theorem for the module of smoothness defined by a nonsymmetric generalized translation operator,” Vestn. MGU, Ser. Mat., (in press). · Zbl 0998.41017 [7] M. K. Potapov and V. M. Fedorov, ”On Jackson’s theorems for a generalized module of smoothness,” Tr. Mat. Inst. Steklova, 172, 291–295 (1985). · Zbl 0575.41005
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.