Completeness of basic trigonometric system in \({\mathcal L}^p\) (Q2782419)

The \(q\)-trigonometric system has been considered in a number of recent papers by S. K. Suslov and others. Some results concerning the completeness of this system have been obtained by \textit{J. Bustoz} and \textit{S. K. Suslov} in [``Basic analog of Fourier series on a \(q\)-quadratic grid''. Methods Appl. Anal. 5, No. 1, 1-38 (1998; Zbl 0961.33013)]. In the paper under review the author extends these results by proving several theorems on the completeness of the \(q\)-trigonometric system in the weighted \(L_{\rho}^{p}\)-spaces for \(1\leq p \leq \infty\), where \(\rho(x)\) is an appropriate weight function. It should be noted that the author uses methods of the theory of entire functions and his results are the \(q\)-analogues of some theorems established by \textit{B. Ja. Levin} [``Distribution of zeros of entire functions'' (1964; Zbl 0152.06703); revised edition (1980)] and \textit{N. Levinson} [``Gap and density theorems'' (1963; Zbl 0145.08003)], regarding the completeness of the classical trigonometric system \(\{e^{i \lambda_{n}x}\}\), where \(\lambda_{n}\) is a suitable sequence of complex numbers. Also, the author gives several examples of complete basic trigonometric systems. The results of this paper are part of a program on the detailed investigation of the theory of \(q\)-Fourier series.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00024].