DOI QR코드

DOI QR Code

UNIVARIATE LEFT FRACTIONAL POLYNOMIAL HIGH ORDER MONOTONE APPROXIMATION

  • Anastassiou, George A. (Department of Mathematical Sciences University of Memphis)
  • Received : 2014.03.21
  • Published : 2015.03.31

Abstract

Let $f{\in}C^r$ ([-1,1]), $r{\geq}0$ and let $L^*$ be a linear left fractional differential operator such that $L^*$ $(f){\geq}0$ throughout [0, 1]. We can find a sequence of polynomials $Q_n$ of degree ${\leq}n$ such that $L^*$ $(Q_n){\geq}0$ over [0, 1], furthermore f is approximated left fractionally and simulta-neously by $Q_n$ on [-1, 1]. The degree of these restricted approximations is given via inequalities using a higher order modulus of smoothness for $f^{(r)}$.

References

  1. G. A. Anastassiou, Bivariate Monotone Approximation, Proc. Amer. Math. 112 (1991), no. 4, 959-964. https://doi.org/10.1090/S0002-9939-1991-1069682-2
  2. G. A. Anastassiou, Higher order monotone approximation with linear differential operators, Indian J. Pure Appl. Math. 24 (1993), no. 4, 263-266.
  3. G. A. Anastassiou and O. Shisha, Monotone approximation with linear differential operators, J. Approx. Theory 44 (1985), no. 4, 391-393. https://doi.org/10.1016/0021-9045(85)90089-9
  4. K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Vol. 2004, 1st edition, Springer, New York, Heidelberg, 2010.
  5. H. H. Gonska and E. Hinnemann, Pointwise estimates for approximation by algebraic polynomials, Acta Math. Hungar. 46 (1985), no. 3-4, 243-254. https://doi.org/10.1007/BF01955736
  6. O. Shisha, Monotone approximation, Pacific J. Math. 15 (1965), 667-671. https://doi.org/10.2140/pjm.1965.15.667