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Systems with nonextendable convergence of quasipolynomials

Authors:

A. A. Ryabinin

Affiliation:

(1) Nizhnii Novgorod State University, USSR

Abstract:

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

$Lleft( z right) = int_{ - a}^a {e^{izt} } dmathcal{L}(t)$

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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