Skip to main content
Log in

Sharp Estimates for Errors of Numerical Differentiation Type Formulas on Trigonometric Polynomials

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

Sharp estimates for errors of formulas of numerical differentiation type on trigonometric polynomials are established. These estimates generalize the well-known Bernshtein-type inequalities. Bibliography: 18 titles.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. A. F. Timan, Approximation of Functions of a Real Variable, International Series of Monographs on Pure and Applied Mathematics, 34, Pergamon Press, Oxford-London-NewYork-Paris (1963).

    Google Scholar 

  2. V. V. Zhuk, Approximation of Periodic Functions, Leningrad (1982).

  3. S. N. Bernshtein, Collections, 1, AN SSSR (1952).

  4. M. Riesz, “Eine Trigonometrische Interpolationsformel und Einige Ungleichungen für Polynome,” Jahresbericht der Deutchen Mathematiker Vereinigung, 23, 354–368 (1914).

    Google Scholar 

  5. R. P. Boas, Jr., “Quelques Généralisations d'un Théorème de S. Bernstein sur la Dérivée d'un Ponynome Trigonométrique,” C. R. Acad. Sci. Paris, 227, 618–619 (1948).

    Google Scholar 

  6. S. N. Bernshtein, “Expansion of S. B. Stechkin's inequality to entire functions of finite order,” Dokl. AN SSSR, 60, No. 9, 1487–1490 (1948).

    Google Scholar 

  7. S. M. Nikol'skii, “Generalization of one S. N. Bernshtein's inequality,” Dokl. AN SSSR, 60, No. 9, 1507–1510 (1948).

    Google Scholar 

  8. S. B. Stechkin, “Generalizaion of some S.N. Bernshtein's inequalities,” Dokl. AN SSSR, 60, No. 9, 1511–1514 (1948).

    Google Scholar 

  9. V. G. Doronin, “Some inequalities for trigonometric polynomials,” Proc. Conf. Approximation Theory and Harmonic Analysis. Tula, 1998, p. 96–97.

  10. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York-London-Toronto (1980).

    Google Scholar 

  11. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, John Wiley and Sons, New York (1972).

    Google Scholar 

  12. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Vol. I, (Bateman Manuscript Project.) McGraw-Hill Book Co., New York-Toronto-London (1953).

    Google Scholar 

  13. G. H. Hardy and W. W. Rogosinski, Fourier Series, Cambridge Tracts. Math. Math. Phys., 38, Cambridge Univ. Press, Cambridge (1956).

    Google Scholar 

  14. P. Civin, “Inequalities for trigonometric integrals,” Duke Math. J., 8, 656–665 (1941).

    Google Scholar 

  15. S. M. Nikol'skii, Approximation of Functions of Many Variables and Embedding Theorems, Moscow, Nauka (1969).

    Google Scholar 

  16. J. F. Steffensen, Interpolation, The William & Wilkins Co., Baltimore (1927).

    Google Scholar 

  17. B. P. Demidovich, Collection of Tasks and Exercises onMathematical Analysis, Nauka, Moscow (1990).

    Google Scholar 

  18. V. V. Zhuk and V. F. Kuzyutin, Approximation of Functions and Numerical Integration, St. Petersburg (1995).

Download references

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vinogradov, O.L., Zhuk, V.V. Sharp Estimates for Errors of Numerical Differentiation Type Formulas on Trigonometric Polynomials. Journal of Mathematical Sciences 105, 2347–2376 (2001). https://doi.org/10.1023/A:1011313229137

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1011313229137

Keywords

Navigation