Univariate simultaneous high order abstract fractional monotone approximation with applications

Abstract

Here we extend our earlier univariate high order simultaneous fractional monotone approximation theory ([3]) to abstract univariate high order simultaneous fractional monotone approximation, with applications to Prabhakar fractional calculus and non-singular kernel fractional calculi. We cover both the left and right sides of this constrained approximation. Let f∈ Cr([- 1 , 1]) , r≥ 0 and let L∗ be a linear abstract left or right fractional differential operator such that L∗(f) ≥ 0 over [0 , 1] or [- 1 , 0] , respectively. We can find a sequence of polynomials Qn of degree ≤ n such that L∗(Qn) ≥ 0 over [0 , 1] or [- 1 , 0] , furthermore f is approximated left or right fractionally and simultaneously by Qn on [- 1 , 1]. The degree of these restricted approximations is given quantitatively by inequalities using a higher order modulus of smoothness for f(r).

Publication Title

Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas

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