Representation of the reciprocal of an entire function by series of partial fractions and exponential approximation

Conditions under which the reciprocal $1/L(\lambda)$ of an entire function with simple zeros $\lambda_k$ can be represented as a series of partial fractions $c_k/(\lambda-\lambda_k)$, $k=1,2,\dots$, are investigated. The possibility of such a representation is characterized, as is conventional, in terms of a particular ‘asymptotically regular’ behaviour of the function $L(\lambda)$. Applications to complete systems of exponentials on a line interval and to representative systems of exponentials in a convex domain are considered.
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