s-Regular functions which preserve a complex slice

Altavilla, Amedeo; Fabritiis, Chiara de

doi:10.1007/s10231-018-0724-1

Cited by 12 publications

(6 citation statements)

References 22 publications

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“…the standard square norm in R n+1 . Moreover we also have that x 2 = x 2 0 − |x| 2 + 2x 0 x and that x 2 = −|x| 2 = −|x c | 2 = (x c ) 2 . Given any nonzero paravector x ∈ R n+1 , its inverse is given by x −1 = x c |x| 2 .…”

Section: Introductionmentioning

confidence: 89%

Implementing zonal harmonics with the Fueter principle

Altavilla,

De Bie,

Wutzig

2019

Preprint

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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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Section: Introductionmentioning

confidence: 89%

Implementing zonal harmonics with the Fueter principle

Altavilla,

De Bie,

Wutzig

2019

Preprint

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“…The set of C J -preserving functions is denoted by R J (Ω), while the set of slice preserving functions by R R (Ω). For a detailed study of the features of slice preserving and one-slice preserving functions, see [2].…”

Section: De Nition 21mentioning

confidence: 99%

Applications of the Sylvester operator in the space of slice semi-regular functions

Altavilla

Fabritiis

2020

Concrete Operators

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In this paper we apply the results obtained in [3] to establish some outcomes of the study of the behaviour of a class of linear operators, which include the Sylvester ones, acting on slice semi-regular functions. We first present a detailed study of the kernel of the linear operator ℒf,g (when not trivial), showing that it has dimension 2 if exactly one between f and g is a zero divisor, and it has dimension 3 if both f and g are zero divisors. Afterwards, we deepen the analysis of the behaviour of the -product, giving a complete classification of the cases when the functions fv, gv and fvgv are linearly dependent and obtaining, as a by-product, a necessary and sufficient condition on the functions f and g in order their *-product is slice-preserving. At last, we give an Embry-type result which classifies the functions f and g such that for any function h commuting with f + g and f * g, we have that h commutes with f and g, too.

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“…it preserves the slice C i . Such kind of functions are called one-slice preserving and are widely studied in [7,8].…”

Section: Prerequisites About Quaternionic Functionsmentioning

confidence: 99%

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

Altavilla¹,

Bisi²

2019

Ann. Acad. Sci. Fenn. Math.

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Given a complex meromorphic function, it is well defined its Riesz measure in terms of the laplacian of the logarithm of its modulus. Moreover, related to this tool, it is possible to prove the celebrated Jensen formula. In the present paper, using among the other things the fundamental solution for the bilaplacian, we introduce a possible generalization of these two concepts in the space of quaternions, obtaining new interesting Riesz measures and global (i.e. four dimensional), Jensen formulas.

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s-Regular functions which preserve a complex slice

Cited by 12 publications

References 22 publications

Implementing zonal harmonics with the Fueter principle

Implementing zonal harmonics with the Fueter principle

Applications of the Sylvester operator in the space of slice semi-regular functions

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

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