Approximation by algebraic polynomials of classes of functions which are fractional integrals of integrable functions (Q1096099)

Approximation in the mean \((E_ n(f)_ 1)\) by algebraic polynomials of order \(\leq n\) is studied in the paper, for classes \(W^ r_ 1\) of functions f, which can be represented as \[ f(x)=(1/\Gamma (r))\int^{1}_{-1}(x-t)_+^{r-1}\phi (t)dt, \] where \(\phi \in L_ 1[-1,1]\), \(\| \phi \|_ 1\geq 1\), \((x-t)_+^{r-1}=[\max (0,x- t)]^{r-1}\), \(\Gamma\) (r) stands for Euler's gamma-function. It is proved that for all real \(r\geq 1\) and positive integers \(n\geq [r]-1\) the relation \(\sup \{E_ n(f)_ 1: f\in W^ r_ 1\}=\| (s_ n)_ r\|_{\infty},\) is valid, where \[ (s_ n)_ r(t)=(1/\Gamma (r))\int^{1}_{-1}(x-t)_+^{r-1} sgn \sin (n+2)arc \cos x dx. \]