# Univariate left general high order fractional monotone approximation

## Abstract

The paper deals with 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 different g. Let f be an r-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 critical closed subinterval I of [a, b]. We can find a sequence of polynomials Qn of degree less-equal n such that L(Qn) is non-negative over I, furthermore, f is fractionally and simultaneously approximated uniformly by Qn over [a, b]. The degree of this constrained approximation is given by inequalities using the high order modulus of smoothness of f(r). We finish with applications of the main fractional monotone approximation theorem for different g.

## Publication Title

Acta Mathematica Universitatis Comenianae

## Recommended Citation

Anastassiou, G.
(2016). Univariate left general high order fractional monotone approximation.* Acta Mathematica Universitatis Comenianae**, 85* (2), 319-335.
Retrieved from https://digitalcommons.memphis.edu/facpubs/6075