On axially rational regular functions and Schur analysis in the Clifford-Appell setting (Q6540301)

The authors are setting up a framework for Schur analysis of quaternionic-valued functions. In complex analysis, a Schur function is a contractive analytic function in the unit disk. Many concepts from scalar complex analysis need to be generalized when the complex numbers are replaced by quaternions from \(\mathbb{H}=\{x = x_0+\sum_{i=1}^3 e_i x_i=x_0+\underline{x}: x_i\in\mathbb{R}\}\). This includes for example Taylor series, the Hardy space \(H_2\) of the unit disk, reproducing kernels, the shift operator, Blaschke products, and (state-space) realizations \(D+zC(I-zA)^{-1}B\) of (rational) functions.\N\NHere the main approach is via axially regular functions that can be written in the form \(f(x)=\sum_{n=0}^\infty \mathcal{Q}_n(z)f_n\) where \(f_n\in\mathbb{H}\) and the \(\mathcal{Q}_n\) are Clifford-Appell polynomials that play the role of monomials in complex analysis. They are obtained as the action of the Fueter map on \(x^n\), \(x\in\mathbb{H}\) (see [\textit{F. Sommen}, J. Math. Anal. Appl. 130, No. 1, 110--133 (1988; Zbl 0634.30042)]). They satisfy for example \(\mathcal{Q}_n \odot_{GCK} \mathcal{Q}_m = \mathcal{Q}_{n+m}\), which requires a special GCK product (the Generalized Cauchy-Kovalevskaya product, see [\textit{A. De Martino} et al., Proc. Edinb. Math. Soc., II. Ser. 66, No. 3, 642--688 (2023; Zbl 1523.30060)]). The Hardy space \(\mathbf{H}_2\) can then be defined as those functions for which \(\sum_{k=0}^\infty|f_n|^2<\infty\) and the reproducing kernels as \(K(z,y)=\sum_{n=0}^\infty \mathcal{Q}_n(x)\overline{\mathcal{Q}_n(y)}\). A Blaschke factor is defined as \(B_a(x)=(1-\mathcal{Q}_1(x)\overline{a})^{-\odot_{GCK}}\odot_{GCK}(a-\mathcal{Q}_1(x))\frac{\overline{a}}{|a|}\), \(a\in\mathbb{B}=\{a\in\mathbb{H}:|a|<1\}\). In a similar way the Schur multipliers and their realizations are defined, in a way that is very similar to the complex case. Throughout the paper, alternative approaches are mentioned and compared, like a hypercomplex approach, using the Fueter mapping theorem, monogenic formulations (using Fueter variables), or the non-generalized CK regular functions. Especially, alternative definitions of a slice hyperholomorphic and a Fueter-map Blaschke factor are worked out.