Скорость рациональн ой аппроксимации и дифференциальные с войства функций

Abstract

The paper deals with the order of best rational approximation of some classes of functions, depending on their differentiability properties. Improvements and generalizations of some results by P. P. Petrushev, V. A. Popov and the author are obtained. The proofs are based on the author's direct rational approximation theorems received recently. One of the results reads as follows. LetR _n (f,L _p ) denote the value of the best approximation of a functionf inL _p ,f∈L _p [0,1], by rational fractions of degree not exceedingn, n≧1. Suppose that 0<p≦∞,s∈NU{0}, andp≠∞ fors=0. Iff is thes-th primitive of some function of bounded variation on [0,1], then

$$\sum\limits_{n = 1}^\infty {\frac{1}{n}(n^{s + 1} R_n (f,L_p ))^2< \infty } $$

.

This statement is exact. Namely, for everys, s∈NU {0}, and every sequence {a _n } ^∞_n=1 ,

$$a_n \geqq a_{n + 1} and \sum n^{ - 1} (n^{s + 1} a_n )^2< \infty ,$$

, there exists a functiong of the classC ^s+1 [0,1] satisfying the inequalities

$$R_n (g, L_p ) \geqq c(p)a_{12} , n = 1, 2, \ldots ,$$

, for everyp, p∈(0, ∞).