We show that properties of pairs of finite, positive, and regular Borel measures on the complex unit circle such as domination, absolute continuity, and singularity can be completely described in terms of containment and intersection of their reproducing kernel Hilbert spaces of “Cauchy transforms” in the complex unit disk. This leads to a new construction of the classical Lebesgue decomposition and proof of the Radon–Nikodym theorem using reproducing kernel theory and functional analysis.

The Fueter-Sce theorem provides a procedure to obtain axially monogenic functions, which are in the kernel of generalized Cauchy–Riemann operator in {\mathbb{R}}^{n+1}${\mathbb{R}}^{n+1}$. This result is obtained by using two operators. The first one is the slice operator, which extends holomorphic functions of one complex variable to slice monogenic functions in \mathbb{R}^{n+1}$\mathbb{R}^{n+1}$. The second one is a suitable power of the Laplace operator in n + 1 variables. Another way to get axially monogenic functions is the generalized Cauchy–Kovalevskaya (CK) extension. This characterizes axial monogenic functions by their restriction to the real line. In this paper, using the connection between the Fueter-Sce map and the generalized CK-extension, we explicitly compute the actions \Delta_{\mathbb{R}^{n+1}}^{\frac{n-1}{2}} x^k$\Delta_{\mathbb{R}^{n+1}}^{\frac{n-1}{2}} x^k$, where x \in \mathbb{R}^{n+1}$x \in \mathbb{R}^{n+1}$. The expressions obtained is related to a well-known class of Clifford–Appell polynomials. These are the building blocks to write a Taylor series for axially monogenic functions. By using the connections between the Fueter-Sce map and the generalized CK extension, we characterize the range and the kernel of the Fueter-Sce map. Furthermore, we focus on studying the Clifford–Appell–Fock space and the Clifford–Appell–Hardy space. Finally, using the polyanalytic Fueter-Sce theorems, we obtain a new family of polyanalytic monogenic polynomials, which extends to higher dimensions the Clifford–Appell polynomials.

In a scale of Fock spaces with radial weights ϕ we study the existence of Riesz bases of (normalized) reproducing kernels. We prove that these spaces possess such bases if and only if ϕ(x) grows at most like (log x)2.

We consider some geometric properties of eigenvectors of linear operators on infinite dimensional Hilbert space. It is proved that the property of a family of vectors ({{x}_{\text{n}}})$({{x}_{\text{n}}})$ to be eigenvectors T{{x}_{n}}={{\lambda }_{n}}{{x}_{n}}({{\lambda }_{n}}\ne {{\lambda }_{k}}\text{for}\,n\ne k)$T{{x}_{n}}={{\lambda }_{n}}{{x}_{n}}({{\lambda }_{n}}\ne {{\lambda }_{k}}\text{for}\,n\ne k)$ of a bounded operator T$T$ (admissibility property) is very instable with respect to additive and linear perturbations. For instance, (1) for the sequence {{\left( {{x}_{n}}+{{\epsilon }_{n}}{{v}_{n}} \right)}_{n\ge k(\epsilon )}}${{\left( {{x}_{n}}+{{\epsilon }_{n}}{{v}_{n}} \right)}_{n\ge k(\epsilon )}}$ to be admissible for every admissible ({{x}_{\text{n}}})$({{x}_{\text{n}}})$ and for a suitable choice of small numbers {{\epsilon }_{n}}\,\ne \,0${{\epsilon }_{n}}\,\ne \,0$ it is necessary and sufficient that the perturbation sequence be eventually scalar: there exist {{\text{ }\!\!\gamma\!\!\text{ }}_{n}}\,\in \,C${{\text{ }\!\!\gamma\!\!\text{ }}_{n}}\,\in \,C$ such that {{v}_{n}}\,=\,{{\gamma }_{n}}{{v}_{k}}\,\text{for}\,n\,\ge \,\text{k}${{v}_{n}}\,=\,{{\gamma }_{n}}{{v}_{k}}\,\text{for}\,n\,\ge \,\text{k}$ (Theorem 2); (2) for a bounded operator A$A$ to transform admissible families ({{x}_{\text{n}}})$({{x}_{\text{n}}})$ into admissible families (A{{x}_{n}})$(A{{x}_{n}})$ it is necessary and sufficient that A$A$ be left invertible (Theorem 4).