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Left general fractional monotone approximation theory

Volume 43 / 2016

George A. Anastassiou Applicationes Mathematicae 43 (2016), 117-131 MSC: 26A33, 41A10, 41A17, 41A25, 41A29. DOI: 10.4064/am2264-12-2015 Published online: 2 December 2015

Abstract

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.

Authors

  • George A. AnastassiouDepartment of Mathematical Sciences
    University of Memphis
    Memphis, TN 38152, U.S.A.
    e-mail

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