Log-biharmonicity and a Jensen formula in the space of quaternions

Altavilla, Amedeo; Bisi, Cinzia

doi:10.5186/aasfm.2019.4447

Cited by 13 publications

(11 citation statements)

References 32 publications

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“…See [11,10,7] for the generalization of the complex Liouville's Theorem to quaternionic functions and slice monogenic functions. The formulas obtained in the present paper have found application also to four dimensional Jensen formulas for quaternionic slice-regular functions [2,21].…”

Section: Introductionmentioning

confidence: 92%

Slice Regularity and Harmonicity on Clifford Algebras

Perotti

2019

Trends in Mathematics

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We present some new relations between the Cauchy-Riemann operator on the real Clifford algebra Rn of signature (0, n) and slice-regular functions on Rn. The class of slice-regular functions, which comprises all polynomials with coefficients on one side, is the base of a recent function theory in several hypercomplex settings, including quaternions and Clifford algebras. In this paper we present formulas, relating the Cauchy-Riemann operator, the spherical Dirac operator, the differential operator characterizing slice regularity, and the spherical derivative of a slice function. The computation of the Laplacian of the spherical derivative of a slice regular function gives a result which implies, in particular, the Fueter-Sce Theorem. In the two four-dimensional cases represented by the paravectors of R3 and by the space of quaternions, these results are related to zonal harmonics on the three-dimensional sphere and to the Poisson kernel of the unit ball of R 4 .

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Section: Introductionmentioning

confidence: 92%

Slice Regularity and Harmonicity on Clifford Algebras

Perotti

2019

Trends in Mathematics

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“…This yields the assertion. The interested reader can find in [4] and in [30] other results about Jensen's Formula but in a slightly different context.…”

Section: Jensen's Formulamentioning

confidence: 99%

The Harmonicity of Slice Regular Functions

Bisi

Winkelmann

2020

J Geom Anal

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In this article, we investigate harmonicity, Laplacians, mean value theorems, and related topics in the context of quaternionic analysis. We observe that a Mean Value Formula for slice regular functions holds true and it is a consequence of the well-known Representation Formula for slice regular functions over $${\mathbb {H}}$$ H . Motivated by this observation, we have constructed three order-two differential operators in the kernel of which slice regular functions are, answering positively to the question: is a slice regular function over $${\mathbb {H}}$$ H (analogous to an holomorphic function over $${\mathbb {C}}$$ C ) ”harmonic” in some sense, i.e., is it in the kernel of some order-two differential operator over $${\mathbb {H}}$$ H ? Finally, some applications are deduced such as a Poisson Formula for slice regular functions over $${\mathbb {H}}$$ H and a Jensen’s Formula for semi-regular ones.

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“…In the recent paper [27], A. Perotti shows some harmonicity properties of slice regular functions, their connections with Clifford analysis and with zonal harmonics in the particular case of real dimension 4. Some of these properties were already exploited in [1] to find a possible generalization of the Jensen formula in higher dimension (see also [28]). Other results connecting the theory of slice regularity with the one of Fueter were given in [26].…”

Section: Harmonicity Of Slice Regular Functionsmentioning

confidence: 99%

“…is homogeneous of degree k. Therefore, thanks to the properties of zonal harmonics, for any integer n ≥ 0 we have that K[∆ 1). The aim of this section is to give an explicit expression of such coefficient in the case in which 1 is replaced by any paravector y such that |y| = 1.…”

Section: A Further Representation Of Zonal Harmonicsmentioning

confidence: 99%

Implementing zonal harmonics with the Fueter principle

Altavilla

Bie

Wutzig

2021

Journal of Mathematical Analysis and Applications

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By exploiting the Fueter theorem, we give new formulas to compute zonal harmonic functions in any dimension. We first give a representation of them as a result of a suitable ladder operator acting on a constant function. Then, inspired by recent work of A. Perotti, using techniques from slice regularity, we derive explicit expressions for zonal harmonics starting from the 2 and 3 dimensional cases. It turns out that all zonal harmonics in any dimension are related to the real part of powers of the standard Hermitian product in C. At the end we compare formulas, obtaining interesting equalities involving the real part of positive and negative powers of the standard Hermitian product.In the two appendices we show how our computations are optimal compared to direct ones.

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Log-biharmonicity and a Jensen formula in the space of quaternions

Cited by 13 publications

References 32 publications

Slice Regularity and Harmonicity on Clifford Algebras

Slice Regularity and Harmonicity on Clifford Algebras

The Harmonicity of Slice Regular Functions

Implementing zonal harmonics with the Fueter principle

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