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 7 publications

(27 citation statements)

References 59 publications

(115 reference statements)

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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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The $$H^\infty $$-Functional Calculi for the Quaternionic Fine Structures of Dirac Type

Colombo,

Pinton,

Schlosser

2024

Milan J. Math.

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In recent works, various integral representations have been proposed for specific sets of functions. These representations are derived from the Fueter–Sce extension theorem, considering all possible factorizations of the Laplace operator in relation to both the Cauchy–Fueter operator (often referred to as the Dirac operator) and its conjugate. The collection of these function spaces, along with their corresponding functional calculi, are called the quaternionic fine structures within the context of the S-spectrum. In this paper, we utilize these integral representations of functions to introduce novel functional calculi tailored for quaternionic operators of sectorial type. Specifically, by leveraging the aforementioned factorization of the Laplace operator, we identify four distinct classes of functions: slice hyperholomorphic functions (leading to the S-functional calculus), axially harmonic functions (leading to the Q-functional calculus), axially polyanalytic functions of order 2 (leading to the $$P_2$$ P 2 -functional calculus), and axially monogenic functions (leading to the F-functional calculus). By applying the respective product rule, we establish the four different $$H^\infty $$ H ∞ -versions of these functional calculi.

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

Cited by 7 publications

References 59 publications

Properties of a polyanalytic functional calculus on the S‐spectrum

Properties of a polyanalytic functional calculus on the S‐spectrum

The Fueter-Sce mapping and the Clifford–Appell polynomials

The $$H^\infty $$-Functional Calculi for the Quaternionic Fine Structures of Dirac Type

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