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An extremal problem for algebraic polynomials with zero mean value on an interval

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Abstract

LetP n O(h) be the set of algebraic polynomials of degreen with real coefficients and with zero mean value (with weighth) on the interval [−1, 1]:

$$\smallint _{ - 1}^1 h(x)p_n (x) dx = 0;$$

hereh is a function which is summable, nonnegative, and nonzero on a set of positive measure on [−1, 1]. We study the problem of the least possible value

$$i_n (h) = \inf \{ \mu (p_n ):p_n \in \mathcal{P}_n^0 \} $$

of the measure μ(P n)=mes{x∈[−1,1]:P n(x)≥0} of the set of points of the interval at which the polynomialp n∈P n O is nonnegative. We find the exact value ofi n(h) under certain restrictions on the weighth. In particular, the Jacobi weight

$$h^{(\alpha ,\beta )} (x) = (1 - x)^\alpha (1 + x)^\beta $$

satisfies these restrictions provided that −1<α, β≤0.

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References

  1. A. G. Babenko,Extremal Properties of Polynomials and Exact Estimates of Mean Quadractic Approximations [in Russian], Kandidat thesis in the physico-mathematical sciences, Sverdlovsk (1987).

  2. S. N. Bernshtein, “On Cotes and Chebyshev quadrature formulas,” in:Collected Works (in 4 vols.) [in Russian], Vol. 2, Moscow (1954), pp. 200–204.

  3. A. G. Babenko, “An extremal problem for polynomials,”Mat. Zametki [Math. Notes],35, No. 3, 349–356 (1984).

    MATH  MathSciNet  Google Scholar 

  4. L. V. Taikov, “A range of extremal problems for trigonometric polynomials,”Uspekhi Mat. Nauk [Russian Math. Surveys],20, No. 3, 205–211 (1965).

    MathSciNet  Google Scholar 

  5. A. G. Babenko, “Jackson’s inequality for mean square approximations of periodic functions by trigonometric polynomial on a uniform mesh,”Mat. Zametki [Math. Notes],43, No. 4, 460–473 (1988).

    MATH  MathSciNet  Google Scholar 

  6. G. Fazekas and V. I. Levinstein, “On upper bounds for code distance and covering radius of designs in polynomial metric spaces,”J. Combin. Theory. Ser. A,70, No. 2, 267–288 (1995).

    MathSciNet  Google Scholar 

  7. V. A. Yudin, “Covering of a sphere and extremal properties of orthogonal polynomials,”Diskret. Mat. [Discrete Math. Appl.],7, No. 3, 81–88 (1995).

    MATH  MathSciNet  Google Scholar 

  8. V. I. Krylov,Approximate Calculation of Integrals [in Russian], Fizmatgiz, Moscow (1959).

    Google Scholar 

  9. G. Szegö,Orthogonal Polynomials, Vol. 2, Amer. Math. Soc., New York (1959).

    Google Scholar 

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Authors and Affiliations

  1. Urals State University, Ekaterinburg

    V. V. Arestov & V. Yu. Raevskaya

Authors
  1. V. V. Arestov
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  2. V. Yu. Raevskaya
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Translated fromMatematicheskie Zametki, Vol. 62, No. 3, pp. 332–342, September, 1997.

Translated by I. P. Zvyagin

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Arestov, V.V., Raevskaya, V.Y. An extremal problem for algebraic polynomials with zero mean value on an interval. Math Notes 62, 278–287 (1997). https://doi.org/10.1007/BF02360868

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  • DOI: https://doi.org/10.1007/BF02360868

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