Interpolation and divisibility of meromorphic matrix functions (Q1916826)

Let \(\Omega\) be a domain of the complex plane and let \(A\) be a discrete subset of \(\Omega\). The problem of interpolating matrix valued meromorphic functions in \(\Omega\) with data (e.g. poles, zeroes, polar parts, divisors, etc) on \(A\) is crucial for numerous applications, see \textit{I. Gohberg} and \textit{L. Rodman}: [J. Anal. Math. 40, 90-128 (1981; Zbl 0496.15013)], and \textit{I. Gohberg} (ed.): Topics interpolation theory of rational matrix-valued functions (1988; Zbl 0651.47008). The present paper deals with the construction of meromorphic matrix functions with given spectral data (null and pole functions, coupling matrix and left annihilator) given on a locally finite covering of the domain \(\Omega\). As an application, the divisibility problem of meromorphic functions is investigated in terms of the same local spectral data. For prior similar results see J. A. Ball, N. Cohen, M. Rakowski and L. Rodman: Spectral data and minimal divisibility of nonregular meromorphic matrix functions, Technical Report 91.04, The College of William and Mary, Williamsburg, VA, 1991.