Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

For citation:

Tyuleneva A. A. Approximation of the Riemann–Liouville Integrals by Algebraic Polynomials on the Segment. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2014, vol. 14, iss. 3, pp. 305-311. DOI: 10.18500/1816-9791-2014-14-3-305-311, EDN: SMSJWF

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
10.09.2014
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Language: 
Russian
Heading: 
Mathematics
UDC: 
517.51
DOI: 
10.18500/1816-9791-2014-14-3-305-311
EDN: 
SMSJWF

Approximation of the Riemann–Liouville Integrals by Algebraic Polynomials on the Segment

Autors: 
Tyuleneva Anna Anotol'evna, Saratov State University
Abstract: 

The direct approximation theorem by algebraic polynomials is proved for Riemann–Liouville integrals of order r>0. As a corollary, we obtain asymptotic equalities for ε-entropy of the image of a Hölder type class under Riemann–Liouville integration operator.

Key words: 
p-variation metric
Lp space
Riemann–Liouville integral
best approximation
algebraic polynomials
ε-entropy
References: 
  1. Terekhin A. P. Approximation of bounded p-variation functions. Izvestiya vuzov. Matematika, 1965, no. 2, pp. 171–187 (in Russian).
  2. Samko S. G., Kilbas A. A., Marichev O. I. Fractional Integrals and Derivatives: Theory and Applications, New York, Gordon and Breach Science, 1993. 1006 p.
  3. Kolmogorov A. N., Tikhomirov V. M. "-entropy and "-capacity of a set in the functional space. Uspehi mat. nauk, 1959, vol. 14, iss. 2, pp. 3–86 (in Russian).
  4. Volosivets S. S. Asymptotic properties of one compact set of smooth functions in the space of functions of bounded p-variation. Math. Notes, 1995, vol. 57, iss. 2, pp. 148–157.
  5. Lorentz G. G. Metric entropy and approximation. Bull. Amer. Math. Soc., 1966, vol. 72, no. 6, pp. 903–927.
  6. Edwards R. Fourier Series: A modern introduction. Vol. 1. New York, Springer, 1982, 234 p.
  7. Ibragimov I. I. On best approximation of a function whose s-th derivative has bounded variation on segment [−1, 1]. Doklady Akad. Nauk SSSR, 1953, vol. 90, no. 1. pp. 13–15 (in Russian).
  8. DeVore R., Lorentz G. G. Constructive approximation. Berlin, Heidelberg, Springer, 1993, 449 p.
  9. Korneichuk N. P. Exact Constants in Approximation Theory, 2009, Cambridge, Cambridge Univ. Press, 2009, 468 p.
  10. Nasibov F. G. On the order of best approximations of functions havong fractional derivative in Riemann – Liouville sense. Izv. AN Azerb. SSR. Ser. fiz.-mat. nauk, 1962, no. 3, pp. 51–57 (in Russian).
  11. Clements G. F. Entropies of several sets of real valued functions. Pacific J. Math., 1963, vol. 13, no. 4, pp. 1085–1095. 
Received: 
19.03.2014
Accepted: 
23.07.2014
Published: 
10.09.2014
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