On the approximation condition of continuity for the fractional derivative

Abstract

For the space $C^\alpha$ of functions having a Marchaud continuous fractional derivative of order $\alpha>0$ on the closed interval $[0, 1]$ and for the function class $H_r [\bar \varepsilon ] = \{ f:E_n (f) \leqslant \varepsilon _n ,n \in N,f^{(i)} (1) = 0, j = \overline {0,r} \}$  it is proved that $H_r [\bar \varepsilon ]\subset C^\alpha$  if and only if  $\sum\limits_{i = 1}^\infty {\varepsilon _i j^{2\alpha - 1}< \infty }$.