Approximation by rational functions in integral metrics and differentiability in the mean

E. P. Dolzhenko ; E. A. Sevast'yanov

Matematičeskie zametki, Tome 16 (1974) no. 5, pp. 801-811

Résumé

The paper deals with approximations of a function $f$ of space $L_p[0,1]$ by rational functions in the metric of this same space ($0<p\le\infty$). It is shown that sufficiently rapid decrease as $n\to\infty$ of the least deviations $R_n(f,р)$ of function$f$ of rational functions of degree no higher than $n$ is evidence of the presence in $f$ of derivatives and differentials of a definite order if differentiation is understood as differentiation in the metric of space $L_q[0,1]$, with $0<q<q(p)$, where $q(p)$ depends on $p$ and the differentiation order, $q(p)<p$.