Left general fractional monotone approximation theory

George A. Anastassiou¹

¹ Department of Mathematical Sciences University of Memphis Memphis, TN 38152, U.S.A.

Applicationes Mathematicae, Tome 43 (2016) no. 1, pp. 117-131

Résumé

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.

Détail

DOI : 10.4064/am2264-12-2015

Keywords: introduce general fractional caputo style derivatives respect absolutely continuous strictly increasing function various examples fractional derivatives different p times continuously differentiable function linear general fractional differential operator non negative closed subinterval sequence polynomials degree non negative furthermore approximated uniformly degree constrained approximation given inequality using first modulus continuity finish applications main fractional monotone approximation theorem different proving main theorem establish useful related general results