Bulletin of the
Korean Mathematical Society BKMS

Let $f\in C^{r}\left( \left[ -1,1\right] \right) $, $r\geq 0$ and let $ L^{\ast }$ be a linear left fractional differential operator such that $ L^{\ast }\left( f\right) \geq 0$ throughout $\left[ 0,1\right] $. We can find a sequence of polynomials $Q_{n}$ of degree $\leq n$ such that $L^{\ast }\left( Q_{n}\right) \geq 0$ over $\left[ 0,1\right] $, furthermore $f$ is approximated left fractionally and simultaneously by $Q_{n}$ on $\left[ -1,1 \right] .$ The degree of these restricted approximations is given via inequalities using a higher order modulus of smoothness for $f^{\left( r\right) }.$

Keywords: monotone approximation, Caputo fractional derivative, fractional linear differential operator, higher order modulus of smoothness

MSC numbers: 26A33, 41A10, 41A17, 41A25, 41A28, 41A29