Complete systems of Hermite associated functions (Q2731889)

It is well known that the system of Hermite polynomials \(\{H_n (z)\}_{n=0}^{\infty}\) is a solution of the linear recurrence equation of second kind \(y_{n+1}-2z y_n + 2n y_{n-1}=0\), \(n = 1,2,3,\dots\) The system of complex functions, defined for \(z \in {\mathbb{C}} \setminus {\mathbb{R}}\) by the equalities NEWLINE\[NEWLINE G_n (z) = - \int_{-\infty}^{\infty} {\frac {\exp (-t^2) H_n (t)} {t-z}} dt, \quad n=0,1,2,\dots, NEWLINE\]NEWLINE and giving another solution of the same equation, is the system of Hermite associated functions. NEWLINENEWLINENEWLINEA criterion for completeness in \({\mathbb{C}}\)-vector spaces \(H(D)\) of complex functions holomorphic in \(D\), due to \textit{B. Ya. Levin} [Distribution of zeros of entire functions (1964; Zbl 0152.06703)], states that for any simply connected region \(D \subset {\mathbb{C}}\), a sequence \(\{f_n (z)\}_{n=0}^{\infty}\subset H(D)\) iff for every rectifiable Jordan curve \(\gamma \subset D\) the only function \(F \in H_{\gamma}\), which is orthogonal on \(\gamma\) to each of the functions \(\{f_n (z)\}_{n=0}^{\infty}\), is identically zero, i.e. the equalities \(\int_{\gamma} f_n (z) F(z) dz = 0\), \(n=0,1,2,\dots\) imply \(F\equiv 0\). NEWLINENEWLINENEWLINEThe main result proved by the author is: Let \(k=\{k_n\}_{n=0}^{\infty}\) be an increasing sequence of positive integers with density greater than \(1/2\), i.e. there exists \(\lim_{n \to \infty} ( n / k_n) = \delta(k) > 1/2\). Then the system \(\{G_{k_n} (z)\}_{n=0}^{\infty}\) is complete in the space \(H(D)\) provided that \(D\) is any simply connected subregion of \({\mathbb{C}}\setminus {\mathbb{R}}\). NEWLINENEWLINENEWLINEThe proof is by assuming the contrary and using the above completeness criterion and a remarkable result of \textit{E. Hille} [Trans. Am. Math. Soc. 47, 80-94 (1940; Zbl 0022.36502)]. It is also remarked that the statement is not true when \(\delta(k)=1/2\).