Analogues of the V. I. Smirnov spaces for nonintegral exponents (Q2638480)

Suppose that \(\Gamma\) is a contour joining points -\(\pi\) and \(\pi\) and that there is \(\theta\in [0,\pi)\) such that every set \(\{x+t \exp (i\theta):\) \(t\in {\mathbb{R}}\}\cap \Gamma\) contains at most one point. Let S be a sine type function (this term denotes a particular kind of entire functions of exponential type) with zeros: \(...,\lambda_{-1},\lambda_ 0,\lambda_ 1,...\), where Re \(\lambda\) \({}_ n\leq Re \lambda_{n+1}\). Let \(\Delta =[-\pi,\pi]\). We define functions \(\psi_ k\in L^ 2(\Delta)\) by \[ \frac{S(\lambda)}{S'(\lambda_ k)(\lambda -\lambda_ k)}=\int^{\pi}_{-\pi}e^{i\lambda t}\psi_ k(t)dt. \] The author proves that the system \(\{\psi_ k\}\) is complete in the space \(L^ 2(\Gamma)\).