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Sharp estimates of best approximations by deviations of Weierstrass-type integrals

We establish the estimates

$$ {A_{\sigma }}{(f)_p}\leq KP\left( {f-f*W} \right), $$

where W is a kernel of special type that is integrable on R and A σ(f) P is the best approximation of a function f with respect to a seminorm P by entire functions of exponential type not greater than σ. For the uniform and integral norms, we find the least possible constant K. The estimates are obtained by linear methods of approximation. Bibliography: 7 titles.

References

  1. N. I. Akhiezer, Lectures on the Theory of Approximation [in Russian], Moscow (1965).

  2. V. V. Zhuk, Approximation of Periodic Functions [in Russian], Leningrad (1982).

  3. O. L. Vinogradov, ``Sharp Jackson-type inequalities for approximations of classes of convolutions by entire functions of exponential type,'' Algebra Analiz, 17, 56--111 (2005).

    Google Scholar 

  4. V. P. Zastavnyi and V. V. Savchuk, ``Approximations of classes of convolutions by linear operators of special form,'' Mat. Zametki, 90, 351--361 (2011).

    MathSciNet  Article  Google Scholar 

  5. N. P. Korneichuk, Exact Constants in Approximation Theory [in Russian], Moscow (1987).

  6. D. V. Widder, The Laplace Transform, Princeton University Press, Princeton (1946).

    Google Scholar 

  7. E. Kamke, The Lebesgue--Stieltjes Integral [Russian translation], Moscow (1959).

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Correspondence to O. L. Vinogradov.

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Translated from Zapiski Nauchnyk Seminarov POMI, Vol. 401, 2012, pp. 53–70.

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Vinogradov, O.L. Sharp estimates of best approximations by deviations of Weierstrass-type integrals. J Math Sci 194, 628–638 (2013). https://doi.org/10.1007/s10958-013-1551-y

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  • DOI: https://doi.org/10.1007/s10958-013-1551-y

Keywords

  • Entire Function
  • Linear Method
  • Exponential Type
  • Sharp Estimate
  • Integral Norm