A note on completeness of basic trigonometric system in \({\mathcal L}^2\) (Q1880859)

Let \(\mathcal {E}_{q}(x;\alpha)\) be the basic exponential function on a \(q\)-quadratic grid and \(J_{\mu}^{(2)}(x;q)\) be Jackson's \(q\)-Bessel function. In a previous paper [Contemp. Math. 291, 229--241 (2001; Zbl 1028.42008)], the author has established the following result: Suppose that \(j_{\mu,\,k}(q)\) are the zeros of \(J_{\mu}^{(2)}(x;q)\). The \(q\)-trigonometric system \(\{\mathcal {E}_{q}(x;i \omega_{k})\}_{k=-\infty}^{\infty}\) is complete in \({\mathcal {L}}^{p}_{\rho}(-1,1)\), where \(1 \leq p<\infty\) and \(\rho(x)\) is an integrable function, positive almost everywhere on \((-1,1)\), for \(0<\nu<1\), if (a) \(\omega_{k}=1/2 j_{\nu-1,\,k}(q)\), \(k=\pm 1, \pm 2,\ldots\); (b) \(\omega_{k}=1/2 j_{\nu,k}(q)\), \(k=0,\pm 1, \pm 2,\ldots\,\). The special case \(\nu=1/2\) is particularly interesting as the corresponding systems are orthogonal. In this paper, the author gives a different proof of the above mentioned result, when \(p=2,\; 0<\nu\leq 1/2\) and \(\rho(x)\) is the weight function for the continuous \(q\)-ultraspherical polynomials. This gives, finally, an independent proof of the completeness of the basic trigonometric system in \({\mathcal {L}}^{2}\). It is worth mentioning that in this new approach the proof uses the \(q\)-Lommel polynomials and several theorems from classical analysis. The special case \(p=2,\;0<\nu<1/2\) and \(\rho(x)\) is the weight function for the continuous \(q\)-ultraspherical polynomials, has been proved by the author in [NATO Sci. Ser. II, Math. Phys. Chem. 30, 411--456 (2001; Zbl 1005.33007)] by a completely different method from the one in the paper under review.