ON CONVERGENCE OF SERIES OF SIMPLE PARTIAL FRACTIONS IN L_P(R)

摘要

We show that the necessary condition for the convergence of the series of simple par-tial fractions ~∞Σ_(k=1) (z-z_k)~(-1) in L_P(R), 1 ＜ p ＜ ∞, is the convergence of the series ~∞Σ_(k=1) |z_k|~(-1/q) ln~(-1-ε) (|z_k|+1), ε ＞ 0. In the case 1 ＜ ρ ＜ 2, we obtain a convergence criterion in terms of the imaginary parts of poles under the condition that all the poles z_k = x_k + i_(yk) belong to the angle |z_k| ＜ α|y_k| with a fixed α ＞ 0. Bibliography. 7 titles.