In this paper we begin the study of Schur analysis and of de Branges–Rovnyak spaces in the framework of Fueter hyperholomorphic functions. The difference with other approaches is that we consider the class of functions spanned by Appell-like polynomials. This approach is very efficient from various points of view, for example in operator theory, and allows us to make connections with the recently developed theory of slice polyanalytic functions. We tackle a number of problems: we describe a Hardy space, Schur multipliers and related results. We also discuss Blaschke functions, Herglotz multipliers and their associated kernels and Hilbert spaces. Finally, we consider the counterpart of the half-space case, and the corresponding Hardy space, Schur multipliers and Carathéodory multipliers.

在本文中，我们开始在 Fueter 超全纯函数的框架内研究 Schur 分析和 de Branges-Rovnyak 空间。与其他方法的不同之处在于我们考虑由类 Appell 多项式跨越的函数类。这种方法从各个角度来看都非常有效，例如在算子理论中，并且允许我们与最近开发的切片多分析函数理论建立联系。我们解决了许多问题：我们描述了 Hardy 空间、Schur 乘子和相关结果。我们还讨论了 Blaschke 函数、Herglotz 乘子及其相关的核和希尔伯特空间。最后，我们考虑半空间情况的对应物，以及相应的 Hardy 空间、Schur 乘子和 Carathéodory 乘子。