Univariate simultaneous high order abstract fractional monotone approximation with applications

Abstract

Here we extend our earlier univariate high order simultaneous fractional monotone approximation theory ([3]) to abstract univariate high order simultaneous fractional monotone approximation, with applications to Prabhakar fractional calculus and non-singular kernel fractional calculi. We cover both the left and right sides of this constrained approximation. Let $f\in C^{r}\left( \left[ -1,1\right] \right)$, $r\ge 0$ and let $L^{*}$ be a linear abstract left or right fractional differential operator such that $L^{*}\left( f\right) \ge 0$ over $\left[ 0,1\right]$ or $\left[ -1,0\right]$, respectively. We can find a sequence of polynomials $Q_{n}$ of degree $\le n$ such that $L^{*}\left( Q_{n}\right) \ge 0$ over $\left[ 0,1\right]$ or $\left[ -1,0\right]$, furthermore f is approximated left or right fractionally and simultaneously by $Q_{n}$ on $\left[ -1,1\right]$. The degree of these restricted approximations is given quantitatively by inequalities using a higher order modulus of smoothness for $f^{\left( r\right) }.$