Gleason's problem and interpolation for hyperholomorphic functions). (Q1565891)

The authors generalize the following Gleason problem: Given a family \({\mathcal F}\) of analytic functions in \(\Omega\subset \mathbb{C}^N\) and given a point \(a\in{\mathcal F}\) such that: \[ f(z)- f(a)= \sum^N_{i=1} (z_i- a_i) g_i(z).\tag{1} \] Let \(N= 2\) and \(\Omega\subset\mathbb{C}^2= \mathbb{H}\). In (1) we consider now functions \(g_i\) which are left hyperholomorhic with quaternionic coefficients. For \(a= 0\) the functions \[ g_i(x):= \int^1_0 \partial_i f(tx)\,dt \] defined in the unit ball \(S\) are also left hyperholomorphic. The proof is obtained using Leibenson's method. For an arbitrary point \(a\in S\) the representation \[ f(x)- (1-\overline a x){(1-| a|^2)^3\over| 1-\overline ax|^4} f(a)= \zeta(x,a) \begin{pmatrix} g_1(x)\\ g_2(x)\\ g_3(x)\end{pmatrix}, \] is obtained where \(\zeta: S\to \mathbb{H}^3\) has to exist and \(g_i\), \(f\) are left hyperholomorphic functions. This result is interpreted in pairs of complex functions. Finally, the homogeneous interpolation problem in the setting of left hyperholomorphic functions can be studied.