Left general fractional monotone approximation theory

We introduce left general fractional Caputo style derivatives with respect to an absolutely continuous strictly increasing function $g$. We give various examples of such fractional derivatives for different $g$. Let $f$ be a $p$-times continuously differentiable function on $[a,b] $, and let $L$ be a linear left general fractional differential operator such that $L(f) $ is non-negative over a closed subinterval $I$ of $[a,b] $. We find a sequence of polynomials $Q_{n}$ of degree $\le n$ such that $L(Q_{n}) $ is non-negative over $I$, and furthermore $f$ is approximated uniformly by $Q_{n}$ over $[a,b].$

The degree of this constrained approximation is given by an inequality using the first modulus of continuity of $f^{(p) }$. We finish with applications of the main fractional monotone approximation theorem for different $g$. On the way to proving the main theorem we establish useful related general results.