Axially harmonic functions and the harmonic functional calculus on the S-spectrum

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

doi:10.48550/arxiv.2205.08162

Cited by 3 publications

(17 citation statements)

References 25 publications

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

Section: Definition 24 (Fueter Regular Functions) Let U ⊂ H Be An Ope...mentioning

confidence: 98%

“…For further information about the S-functional calculus see [18,22], whereas for the F -functional calculus see [14,18,19]. The properties of the Q-functional calculus are studied in [15].…”

Section: O(d)mentioning

confidence: 99%

“…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 D to the slice Cauchy kernel you get the pseudo Cauchy kernel (see [15]).…”

Section: Definition 24 (Fueter Regular Functions) Let U ⊂ H Be An Ope...mentioning

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: 2mentioning

confidence: 99%

“…The polyanalytic functional calculus of order 2 on the S-spectrum was introduced in [26]. Similarly as the harmonic functional calculus (see [15]), it can be seen as an intermediate functional calculus between the S-functional calculus and the F -functional calculus (see [18]).…”

Section: Introductionmentioning

confidence: 99%

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A polyanalytic functional calculus of order 2 on the 𝑆-spectrum

Martino

Pinton

2023

Proc. Amer. Math. Soc.

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The Fueter mapping theorem gives a constructive way to extend holomorphic functions of one complex variable to monogenic functions, i.e., null solutions of the generalized Cauchy-Riemann operator in R 4 , denoted by D. 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 these 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 ∆, to a slice hyperholomorphic function. The polyanalytic functional calculus, that we study in this paper, is based on the factorization of ∆ = DD. Instead of applying directly the Laplace operator to a slice hyperholomorphic function we apply first the operator D and we get a polyanalytic function of order 2, i..e, a function that belongs to the kernel of D 2 . 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 D to the Cauchy formulas in Theorem 2.13 we the following result(see [15]).…”

Section: Definition 24 (Fueter Regular Functions) Let U ⊂ H Be An Ope...mentioning

confidence: 98%

Section: O(d)mentioning

confidence: 99%

Section: Definition 24 (Fueter Regular Functions) Let U ⊂ H Be An Ope...mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A polyanalytic functional calculus of order 2 on the 𝑆-spectrum

Martino

Pinton

2023

Proc. Amer. Math. Soc.

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

et al. 2022

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The spectral theory on the S-spectrum was introduced to give an appropriate mathematical setting to quaternionic quantum mechanics, but it was soon realized that there were different applications of this theory, for example, to fractional heat diffusion and to the spectral theory for the Dirac operator on manifolds. In this seminal paper we introduce the harmonic functional calculus based on the S-spectrum and on an integral representation of axially harmonic functions. This calculus can be seen as a bridge between harmonic analysis and the spectral theory. The resolvent operator of the harmonic functional calculus is the commutative version of the pseudo S-resolvent operator. This new calculus also appears, in a natural way, in the product rule for the F-functional calculus.

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Generalized Appell polynomials and Fueter–Bargmann transforms in the polyanalytic setting

Martino

Diki

2022

Anal. Appl.

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This paper deals with some special integral transforms in the setting of quaternionic valued slice polyanalytic functions. In particular, using the polyanalytic Fueter mappings, it is possible to construct a new family of polynomials which are called the generalized Appell polynomials. Furthermore, the range of the polyanalytic Fueter mappings on two different polyanalytic Fock spaces is characterized. Finally, we study the polyanalytic Fueter–Bargmann transforms.

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Axially harmonic functions and the harmonic functional calculus on the S-spectrum

Cited by 3 publications

References 25 publications

A polyanalytic functional calculus of order 2 on the 𝑆-spectrum

A polyanalytic functional calculus of order 2 on the 𝑆-spectrum

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

Generalized Appell polynomials and Fueter–Bargmann transforms in the polyanalytic setting

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