Convergence of Fourier series on the systems of rational functions on the real axis (Q283150)

The author considers a system of functions \(\{\Phi_{n}(z)\}_{n}\), \(n\in N\), and proves that it is orthonormalized on the real axis \(R\), where \(\Phi_{n}(z)=\Phi_{n}^{+}(z)\) for \( n=0,1,\dots\) and \(\Phi_{n}(z)=\Phi_{-n}^{-}(z)\) for \(n=-1,-2,\dots\) and \(\Phi_{n}^{+}(z)\), \(\Phi_{-n}^{-}(z)\) are defined by the set of points \(a=\{ \alpha_{k}\}_{k=0}^{\infty}\) with \(\mathrm{Im } a_{k}>0\) and \(b=\{ \beta_{k}\}_{k=0}^{\infty}\) with \(\mathrm{Im } b_{k}<0\). The main result is that for any function \(f\in L_{p}(R)\), \(p>1\), its Fourier series in the system \(\Phi_{n}(z)\) converges to this function in the metric of the space \(L_{p}(R)\). For the proof, the author takes the partial sums of these series and then he obtains a convenient integral representation of them.