Harmonic Conjugates in Weighted Bergman Spaces of Quaternion-Valued Functions

Avetisyan, Karen; Gürlebeck, Klaus; Sprößig, Wolfgang

doi:10.1007/bf03321747

Cited by 8 publications

(10 citation statements)

References 18 publications

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“…The space of all harmonic functions h.B n ; R/ with kuk p;˛< 1 for u 2 h.B n ; R/ is denoted by hp. In Avetisyan et al (2009) A corresponding result also exists for 0 < p < 1. The proof is based on the weighted Bergman kernel…”

Section: Harmonic Conjugation In Bergman Spacementioning

confidence: 63%

“…In 1979 it was Sudbery who constructed explicitly a quaternionic-valued monogenic function whose scalar part is a given real-valued harmonic function. Much later in Avetisyan et al (2009), it is shown that if the given harmonic function belongs to a special Bergman space, then also the vectorial part (the conjugate harmonic function) and the whole constructed monogenic function belong to the same Bergman space. The weighted Bergman norm is introduced by…”

Section: Poisson's Formulamentioning

confidence: 98%

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Clifford Analysis and Harmonic Polynomials

Gürlebeck

Sprößig

2013

Handbook of Geomathematics

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This overview gives an insight in the new field of hypercomplex analysis in relation to harmonic analysis. The algebra of complex numbers is replaced by the non-commutative algebra of real quaternions or by Clifford algebras. This contribution is focused on the presentation of an operator calculus on the sphere as well as the discussion of monogenic and holomorphic orthonormal systems of polynomials and special functions on the unit ball and the sphere. Function theoretic elements are lined out. Relations to the Lie algebra SO.3/ are discussed and a corresponding Radon transform is introduced.

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Section: Harmonic Conjugation In Bergman Spacementioning

confidence: 63%

Section: Poisson's Formulamentioning

confidence: 98%

Clifford Analysis and Harmonic Polynomials

Gürlebeck

Sprößig

2013

Handbook of Geomathematics

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“…Moreover, the inequality is also valid for 0 < p < 1, but we do not consider this case in the present paper, cf. ,Lemma 4.□…”

Section: Harmonic Conjugates In Weighted Monogenic Bergman Spacesmentioning

confidence: 94%

On Riesz systems of harmonic conjugates in

Morais

Avetisyan

Gürlebeck

2013

Math Methods in App Sciences

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In continuation of recent studies, we discuss two constructive approaches for the generation of harmonic conjugates to find null solutions to the Riesz system in double-struckR3. This class of solutions coincides with the subclass of monogenic functions with values in the reduced quaternions. Our first algorithm for harmonic conjugates is based on special systems of homogeneous harmonic and monogenic polynomials, whereas the second one is presented by means of an integral representation. Some examples of function spaces illustrating the techniques involved are given. More specifically, we discuss the (monogenic) Hardy and weighted Bergman spaces on the unit ball in double-struckR3 consisting of functions with values in the reduced quaternions. We end up proving the boundedness of the underlying harmonic conjugation operators in certain weighted spaces. Copyright © 2013 John Wiley & Sons, Ltd.

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“…A thorough treatment is listed in the bibliography, e.g. Sudbery [14], Xu [15], Brackx, Delanghe and Sommen [2], Brackx and Delanghe [3], Avetisyan, Gürlebeck and Sprößig [1], and Morais et al [6,7,9]. The main point in the approach presented in [2,3] as well as Sudbery's formula [14] is the construction of harmonic conjugates in R 4 "function by function".…”

Section: Generation Of a -Valued Monogenic Functions By Conjugate Harmentioning

confidence: 99%

“…So far no effort has been made to the question to which function spaces these conjugate harmonics and the whole monogenic function belong. In [1] this question was studied for conjugate harmonics via Sudbery's formula in the scale of Bergman spaces. These results are, however, not applicable to functions with values in the reduced quaternions.…”

Section: Generation Of a -Valued Monogenic Functions By Conjugate Harmentioning

confidence: 99%

On 3D Riesz systems of harmonic conjugates

Avetisyan¹,

Gürlebeck²,

Morais³

2012

AIP Conference Proceedings

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This note announces some results that will be presented in the forthcoming paper [10]. In continuation to these studies we discuss a constructive approach for the generation of harmonic conjugates to find nullsolutions to the Riesz system in R 3 . This class of solutions coincides with the subclass of monogenic functions with values in the reduced quaternions. The algorithm for harmonic conjugates is presented by means of an integral representation. Additionally, we discuss the weighted (monogenic) Hardy and Bergman spaces on the unit ball in R 3 consisting of functions with values in the reduced quaternions. We end up showing the boundedness of the underlying harmonic conjugation operators in certain weighted spaces.

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Harmonic Conjugates in Weighted Bergman Spaces of Quaternion-Valued Functions

Cited by 8 publications

References 18 publications

Clifford Analysis and Harmonic Polynomials

Clifford Analysis and Harmonic Polynomials

On Riesz systems of harmonic conjugates in

On 3D Riesz systems of harmonic conjugates

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