Axially Harmonic Functions and the Harmonic Functional Calculus on the S-spectrum

Colombo, Fabrizio; Martino, Antonino De; Pinton, Stefano; Sabadini, Irene

doi:10.1007/s12220-022-01062-3

Cited by 9 publications

(27 citation statements)

References 59 publications

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“…By applying the Fueter operator  to the Cauchy formulas in Theorem 2.13, we obtain the following result (see [15]).…”

Section: Hyperholomorphic Functionsmentioning

confidence: 95%

“…The main advantage to have the Fueter mapping theorem in integral form is that it is possible to obtain a monogenic function by computing the integral of a suitable slice hyperholomorphic function. By applying only the Fueter operator  to the second form of the slice Cauchy kernel, one gets the pseudo Cauchy kernel (see [15]).…”

Section: Hyperholomorphic Functionsmentioning

confidence: 99%

“…We conclude this section with the definition of the 𝑄-functional calculus (harmonic functional calculus). This is crucial to get a product rule for the 𝐹-functional calculus (see [15]). Definition 2.30 (𝑄-functional calculus on the 𝑆-spectrum).…”

Section: Definition 227 (𝐹-Resolvent Operators) Let 𝑇 ∈ (𝑋)mentioning

confidence: 99%

“…For further information about the 𝑆-functional calculus, see [18,23], whereas for the 𝐹-functional calculus, see [14,18,20]. The properties of the 𝑄-functional calculus are studied in [15].…”

Section: Introductionmentioning

confidence: 99%

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Properties of a polyanalytic functional calculus on the S‐spectrum

De Martino,

Pinton

2023

Mathematische Nachrichten

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The Fueter mapping theorem gives a constructive way to extend holomorphic functions of one complex variable to monogenic functions, that is, null solutions of the generalized Cauchy–Riemann operator in , denoted by . This theorem is divided in two steps. In the first step, a holomorphic function is extended to a slice hyperholomorphic function. The Cauchy formula for this type of functions is the starting point of the S‐functional calculus. In the second step, a monogenic function is obtained by applying the Laplace operator in four real variables, namely, Δ, to a slice hyperholomorphic function. The polyanalytic functional calculus, that we study in this paper, is based on the factorization of . Instead of applying directly the Laplace operator to a slice hyperholomorphic function, we apply first the operator and we get a polyanalytic function of order 2, that is, a function that belongs to the kernel of . We can represent this type of functions in an integral form and then we can define the polyanalytic functional calculus on S‐spectrum. The main goal of this paper is to show the principal properties of this functional calculus. In particular, we study a resolvent equation suitable for proving a product rule and generate the Riesz projectors.

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“…By applying the Fueter operator  to the Cauchy formulas in Theorem 2.13, we obtain the following result (see [15]).…”

Section: Hyperholomorphic Functionsmentioning

confidence: 95%

Section: Hyperholomorphic Functionsmentioning

confidence: 99%

Section: Definition 227 (𝐹-Resolvent Operators) Let 𝑇 ∈ (𝑋)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Properties of a polyanalytic functional calculus on the S‐spectrum

De Martino,

Pinton

2023

Mathematische Nachrichten

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“…We used the Clifford–Appell approach for the proof of Theorem 4.11 since it will be of interest in the sequel and it will be also useful to characterize the functions that belong to the kernel of the operator (see [17]). However, it is possible to show this result using Theorem 2.15.…”

Section: The Kernel and Range Of Fueter-sce Mappingmentioning

confidence: 99%

The Fueter-Sce mapping and the Clifford–Appell polynomials

Martino¹,

Diki²,

Adán³

2023

Proceedings of the Edinburgh Mathematical Society

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The Fueter-Sce theorem provides a procedure to obtain axially monogenic functions, which are in the kernel of generalized Cauchy–Riemann operator in ${\mathbb{R}}^{n+1}$ . 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}$ . 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 $\Delta_{\mathbb{R}^{n+1}}^{\frac{n-1}{2}} x^k$ , where $x \in \mathbb{R}^{n+1}$ . 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.

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