A necessary condition for the convergence of simple partial fractions in \( L_p(\mathbb R)\)
Let \(p>1\) and let \((z_k)\) be a sequence of complex numbers having non-vanishing imaginary part. In the main theorem, the author shows that the condition \[ \sum_{k=1}^\infty \frac{k^{p-1}}{|z_k^*|^{p-1}} < \infty, \] where \((z_k^*)\) is the sequence obtained from \((z_k)\) by arranging \(|z_k|\) in increasing order, is necessary for the convergence of the series \(\sum_{k=1}^\infty 1/(t-z_k)\) of simple partial fractions in \(L_p(\mathbb{R})\).
scientific article; zbMATH DE number 6138371