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 Qₙ of degree ≤n such that L(Qₙ) is non-negative over I, and furthermore f is approximated uniformly by Qₙ over [a,b]. The degree of this constrained approximation is given by an inequality using the first modulus of continuity of . 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-am2264-12-2015,
author = {George A. Anastassiou},
title = {Left general fractional monotone approximation theory},
journal = {Applicationes Mathematicae},
volume = {43},
year = {2016},
pages = {117-131},
zbl = {1342.26019},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-am2264-12-2015}
}
George A. Anastassiou. Left general fractional monotone approximation theory. Applicationes Mathematicae, Tome 43 (2016) pp. 117-131. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-am2264-12-2015/