Given Modulus Of Continuity Research Articles

On rational approximation of convex functions with a given modulus of continuity

Доказывается, что для наименьших равномер ных рациональных уклоне нийRn(f) выпуклой на [0,1] функции с модулем непрерывно сти, не превосходящемω(δ), сп раведлива оценка $$R_n (f) \leqq c\frac{{\ln ^2 n}}{{n^2 }}\mathop {\max }\limits_{e^{ - n} \leqq \theta< 1} \left\{ {\omega (\theta )\ln \frac{1}{\theta }} \right\},$$ гдес — абсолютная по стоянная.

APPROXIMATION, BY RATIONAL FUNCTIONS, OF CONVEX FUNCTIONS WITH GIVEN MODULUS OF CONTINUITY

We denote by the least deviation of the continuous function , , from the rational functions of order at most .We establish the following theorems.Theorem 1. Let be convex on () with modulus of continuity . Then where is an absolute constant.Theorem 2. There exist a convex function and a sequence such that 1) , , and 2) , where is an absolute constant.Bibliography: 8 titles.

RATIONAL APPROXIMATIONS TO CONVEX FUNCTIONS WITH GIVEN MODULUS OF CONTINUITY

It is shown that for any convex continuous functions (, ) with modulus of continuity the order of approximation by rational functions does not exceed where is an absolute constant and .Bibliography: 6 items.