UNIVARIATE LEFT FRACTIONAL POLYNOMIAL HIGH ORDER MONOTONE APPROXIMATION

Abstract

Abstract. Let f ∈ C r ([−1,1]), r ≥ 0 and let L ∗ be a linear left frac-tional differential operator such that L ∗ (f) ≥ 0 throughout [0,1]. We canfind a sequence of polynomials Q n of degree ≤ nsuch that L ∗ (Q n ) ≥ 0over [0,1], furthermore f is approximated left fractionally and simulta-neously by Q n on [−1,1].The degree of these restricted approximationsis given via inequalities using a higher order modulus of smoothness forf (r) . 1. IntroductionThe topic of monotone approximation started in [6] has become a majortrend in approximation theory. A typical problem in this subject is: given apositive integer k, approximate a given function whose kth derivative is ≥ 0 bypolynomials having this property.In [3] the authors replaced the kth derivative with a linear differential oper-ator of order k. We mention this motivating result.Theorem 1. Let h,k,p be integers, 0 ≤ h ≤ k ≤ p and let f be a real function,f (p) continuous in [−1,1] with modulus of continuity ω 1 f (p) ,xthere. Leta j (x), j = h,h + 1,...,k be real functions, defined and bounded on [−1,1]and assume a