Interpolation formulas for harmonic functions (Q1288266)

It is known that a real-valued entire harmonic function \(u\) of exponential type less than \(\pi\) is uniquely determined by its values at the points \(n\) and \(ne^{i\alpha}\), \(n = 0,\pm 1,\pm 2,\dots\), unless \(\alpha\) is a rational multiple of \(\pi\). For \(\alpha = \pi/2\), which belongs to the exceptional cases, Ching has proved that \(u\) is uniquely determined by its values at these points if \(u\) is in addition an odd function. In the present paper we shall extend this result to the case \(\alpha = (2k + 1)\pi/(2l)\), where \(k\) and \(l\neq 0\) are arbitrary integers. Furthermore, we shall present formulas which allow a reconstruction of real-valued entire harmonic functions of exponential type \(\pi\) by their samples at the points \(n\) and \(ne^{i\alpha}\), \(n = 0,\pm 1,\pm 2,\dots,\) when \(\alpha = (2k + 1)\pi/(2l)\) or when \(\alpha/\pi\) is irrational and algebraic.