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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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