Spline left fractional monotone approximation involving left fractional differential operators

Abstract

Let f ∈ C^s ([−1, 1]), s∈ IN and L^∗ be a linear left fractional differential operator such that L^∗(f) ≥ 0 on [0,1]. Then there exists a sequence Q_n, n ∈ IN of polynomial splines with equally spaced knots of given fixed order such that L^∗ (Q_n) ≥ 0 on [0, 1]. Furthermore f is approximated with rates fractionally and simultaneously by Q_n in the uniform norm. This constrained fractional approximation on [−1, 1] is given via inequalities invoving a higher modulus of smoothness of f^(s).