Systems with nonextendable convergence of quasipolynomials

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.