Complete Fractional Monotone Approximation

Abstract

Here is developed the theory of complete fractional simultaneous monotoneuniform polynomial approximation with rates using mixed fractional lin-ear di¤erential operators.To achieve that, we establish rst ordinary simultaneous polynomialapproximation with respect to the highest order right and left fractionalderivatives of the function under approximation using their moduli ofcontinuity. Then we derive the complete right and left fractional simulta-neous polynomial approximation with rates, as well we treat their a¢ necombination. Based on the last and elegant analytical techniques, we de-rive preservation of monotonicity by mixed fractional linear di¤erentialoperators. We study special cases.