Abstract Fractional Monotone Approximation with Applications

Here we extended our earlier fractional monotone approximation theory to abstract 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∈Cp−1,1, p≥0 and let L be a linear abstract left or right fractional differential operator such that Lf≥0 over 0,1 or −1,0, respectively. We can find a sequence of polynomials Qn of degree ≤n such that LQn≥0 over 0,1 or −1,0, respectively. Additionally f is approximated quantitatively with rates uniformly by Qn with the use of first modulus of continuity of fp.

Constrained Approximation with Jacobi Weights

In this paper, we prove that for ℓ = 1 or 2 the rate of best ℓ- monotone polynomial approximation in the Lp norm (1 ≤ p ≤) weighted by the Jacobi weight with , is bounded by an appropriate (ℓ + 1)-st modulus of smoothness with the same weight, and that this rate cannot be bounded by the (ℓ + 2)-nd modulus. Related results on constrained weighted spline approximation and applications of our estimates are also given.