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Systems with nonextendable convergence of quasipolynomials

Authors:

A A Ryabinin

Institution:

(1) Nizhnii Novgorod State University, USSR

Abstract:

The system $e \left( \Lambda \right) = \left\{ {(it)^k e^{i\lambda _n t} , 0 \leqslant k \leqslant m_n - 1} \right\}_{n = 1}^\infty$ , where Λ={λ_n} is the set of zeros (of multiplicitiesm _n) of the Fourier transform

$L\left( z \right) = \int_{ - a}^a {e^{izt} } d\mathcal{L}(t)$

of a singular Cantor-Lebesgue measure, is examined. We prove thate(Λ) is complete and minimal inL _p (−a, a) withp≥1, and that |L(x+iy)|² does not satisfy the Muckenhoupt condition on any horizontal line Imz=y≠0 in the complex plane. This implies thate(Λ) does not have the property of convergence extension. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 728–733, November, 1998.

Keywords:

convergence of quasipolynomials convergence extension to the right and left singular Cantor-Lebesgue measure complete minimal system of functions MuckenhouptA ₂ condition sequence of zeros of an entire function

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