Interpolation by splines on finite and infinite planar sets (Q1071894)

Let q be a complex number, \(0<| q| <\infty\). \(\Gamma\) denotes the planar curve \(Z=q^ x\), \(-\infty <x<\infty\). The author wants to find splines on \(\Gamma\) which interpolate on the set \(\{q^ v\}^{k_ 2}_{k_ 1}\), where \(k_ 1,k_ 2\) may be finite integers, or \(k_ 1=-\infty\), \(k_ 2=+\infty.\) For the cases q real, \(k_ 1=-\infty\), \(k_ 2=+\infty\), or \(| q| =1\), many authors have dealt with this problem. But if q is an arbitrary complex number and the number of interpolating points is finite or infinite, which classes of splines could be possible? In the first part of this paper, the author introduces several classes of splines which interpolate on a finite point set. The second part deals with interpolation by splines on infinite sets of data points.