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Conditions for the completeness of the system of polynomials

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Abstract

We consider the space A2(K,γ) of functions which are analytic in the unit disc K and squaresummable in K with respect to plane Lebesgue measureσ with weightγ=¦D¦2, D∈ A2(K, 1), D(z) ≠ 0, z ∈ K. We establish the inequality

$$\smallint _K |Dg|^2 u d\sigma \leqslant \smallint _K u d\sigma ,$$

where g represents the distance from 1/D to the closure of the polynomials [in the metric of A2(K,γ)] and u is any function which is harmonic and nonnegative in K. By means of this inequality we obtain sufficient conditions for the completeness of the system of polynomials in A2(K,γ) in terms of membership of certain functions of D in the class H2 (Hardy-2).

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Literature cited

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Authors and Affiliations

  1. New Moscow Branch of the Moscow Chemicotechnological Institute, USSR

    F. S. Lisin

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  1. F. S. Lisin
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Translated from Matematicheskie Zametki, Vol. 18, No. 4, pp. 507–513, October, 1975.

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Lisin, F.S. Conditions for the completeness of the system of polynomials. Mathematical Notes of the Academy of Sciences of the USSR 18, 891–894 (1975). https://doi.org/10.1007/BF01153040

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  • DOI: https://doi.org/10.1007/BF01153040

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