Representation of functions in the unit disk by series of rational fractions

T. A. Leont'eva

Sbornik. Mathematics, Tome 13 (1971) no. 2, pp. 309-322

Résumé

It is shown that if $f(z)=\sum_{n=0}^\infty a_nz^n$, $a_n=O(1/n^p)$, $p>1$, then $f(z)$ can be expanded in a series $$ f(z)=\sum_{k=1}^\infty\frac{A_k}{1-\lambda_kz},\qquad|\lambda_k|1, $$ that converges uniformly inside the unit disk $|z|1$. For $p>2$ the expansion is valid in the closed disk $|z|\leqslant1$, and $\sum_{k=1}^\infty|A_k|\infty$. Bibliography: 6 titles.