The Hilbert space basis and Hilbert's eighth problem

The paper considers the Hilbert space ${\widehat{H}}_r$ of real functions summable with the square $L^2(a,b{)}_r$ on any interval ${(a,b)r}{r=1}^{\infty}\in $\mathbb{R}$$. It is shown on the basis of the theorem on zeros of real orthogonal polynomials if in ${\widehat{H}}_r$ there exists a complete orthonormal basis $\{f(x{)}_k{\}}_{k=1}^{\mathrm{\infty }}$ and the function $f(x)\in \{f(x{)}_k{\}}_{k=1}^{\mathrm{\infty }}$ has zeros, then these zeros are simple and real. The generalized Hardy function $Z(\sigma ,t)=\mathrm{\Re }\zeta (\sigma +it)e^{i\theta (t)}$ is considered. It is shown that in the Hilbert space ${\widehat{H}}_r$ there exists a complete basis $\{Z({\lambda }_k,t{\}}_{k=1}^{\mathrm{\infty }}$ where ${\lambda }_k\in \mathbb{Q}$ and $Z(t)\in \{Z({\lambda }_k,t{\}}_{k=1}^{\mathrm{\infty }}$ when ${\lambda }_k=1\mathrm{/}2$, hence the Hardy function $Z(t)=\zeta (1\mathrm{/}2+it)e^{i\theta (t)}$ has all simple and real zeros.