Complete Fractional Monotone Approximation
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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.
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Anastassiou, G.
(2015).
Complete Fractional Monotone Approximation.
Acta Mathematica Universitatis Comenianae, 84(1), 103-121.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/69/130
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References
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[2] G.A. Anastassiou, O. Shisha, Monotone approximation with linear di¤eren-
tial operators, J. Approx. Theory 44 (1985), 391-393.
[3] K. Diethelm, The Analysis of Fractional Di¤erential Equations, Lecture
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[4] A.M.A. El-Sayed, M. Gaber, On the nite Caputo and nite Riesz derivatives, Electronic Journal of Theoretical Physics, Vol. 3, No. 12 (2006), 81-95.
[5] O. Shisha, Monotone approximation, Paci c J. Math. 15 (1965), 667-671.
[2] G.A. Anastassiou, O. Shisha, Monotone approximation with linear di¤eren-
tial operators, J. Approx. Theory 44 (1985), 391-393.
[3] K. Diethelm, The Analysis of Fractional Di¤erential Equations, Lecture
Notes in Mathematics, Vol. 2004, 1st edition, Springer, New York, Heidelberg, 2010.
[4] A.M.A. El-Sayed, M. Gaber, On the nite Caputo and nite Riesz derivatives, Electronic Journal of Theoretical Physics, Vol. 3, No. 12 (2006), 81-95.
[5] O. Shisha, Monotone approximation, Paci c J. Math. 15 (1965), 667-671.