ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 41 Motornaya, Oxana, and Motornyi, Vitaly, Dnepropetrovsk State University, Ukraine On the Approximation of the Functions by Algebraic Polynomials Our report contains some recent results on the asymptotic behavior of the best L_{1}-approximation of the certain classes of the functions by algebraic polynomials. Also we will touch some questions on the pointwise approximation by algebraic polynomials of the functions which are fractional integrals of bounded functions. Let W^{r}, r > 0, be the class of functions f represented in the form

f(x) = \frac{1}{Γ(r)} ∫_{−1}^{1}(x − t)_{+}^{r−1}ϕ(t)dt + P (x),

where Γ(r) is the Euler gamma function, |ϕ(t)| ≤ 1 almost everywhere. Then for any f ∈ W^{r} the inequality

|f(x) − P_{n}(x)| ≤ \frac{K_{r}}{n^{r}} (\sqrt{1 − x^{2}})^{r} + O (\frac{ln n}{n^{r+1}} (\sqrt{1 − x^{2}} + 1/n)^{r−1})

holds, where r is an integer and K_{r} is Favard constant.

1. Motornyi V. P. and Motornaya O. V., The best approximation of the classes of differentiable functions by algebraic polynomials in the average, Proceedings of the Steklov Institute of Mathematics 210 (1995), 171–188.

2. Motornyi V. P. and Motornaya O. V. On the best approximation of the classes W^{r}H^{α} by algebraic polynomials in L_{1}, East J. Approx., 1995, 1(1995), 309–339.