A regularization method for a class of dual series equations (Q2784880)

The purpose of this paper is to construct a method of regularization to the dual series equations \(\sum_{n=-\infty}^{+\infty}n x_n e^{i n \varphi} - a\sum_{n=-\infty}^{+\infty}n x_n e^{i n \varphi} = \sum_{n=-\infty}^{+\infty} g_n e^{i n \varphi},\) \(|\varphi|<\theta,\) where \(g_n = f_n+\sum_{m=-\infty}^{+\infty}V_{nm} x_m,\) \(x_n\in l_2(1) = \{x = (x_n)_{n=-\infty}^{+\infty}: \sum_{n=-\infty}^{+\infty}|x_n|^2(|n|+1)<\infty\}.\) And there is additionally supposed that \( f = (f_n)_{n=-\infty}^{+\infty} \in l_2(1), \) the matrix operator \( V: l_2 \to l_2 \) is bounded, series \(\sum_{n=-\infty}^{+\infty} x_n e^{i n \varphi}\) are the Fourier expansions for a function from \( L_2 [-\pi, \pi],\) and \(\sum_{n = -\infty}^{-1} n x_n e^{i n \varphi},\) \(\sum_{n=1}^{\infty} n x_n e^{i n \varphi},\) \(\sum_{n=-\infty}^{+\infty} g_n e^{i n \varphi} \) are the Fourier expansions for a function from \( L_1[-\pi, \pi].\) NEWLINENEWLINENEWLINEThe main idea of the paper is to reduce the original problem to the Riemann-Hilbert problem for an unclosed contour of the unit circle. So the original problem is reduced to the operator equation \(x+Hx=h,\) where \(H=AV:l_2\to l_2, \) is a compact operator and \(h=Af\).