Univariate Left General High order Fractional Monotone Approximation

Abstract

Here are used the left general fractional derivatives Caputo style with respect to a base absolutely continuous strictly increasing function g. We mention various examples of such fractional derivatives for di¤erent g. Let f be r-times continuously di¤erentiable function on [a; b], and let L be a linear left general fractional di¤erential operator such that L(f) is non-negative over a critical closed subinterval I of [a; b]. We can find a sequence of polynomials Q_n of degree less-equal n such that L(Q_n) is non-negative over I, furthermore f is fractionally and simultaneously approximated uniformly by Q_n over [a; b].The degree of this constrained approximation is given by inequalities using the high order modulus of smoothness of f(r). We nish with applications of the main fractional monotone approximation theorem for different g.