Hypercomplex Fourier and Laplace transforms I

Sommen, Franciscus

doi:10.1215/ijm/1256046802

Cited by 30 publications

(22 citation statements)

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“…Quaternion-valued integral transforms in the context of quaternionic analysis nowadays extend analytic tools in mathematical physics and engineering applications (see, e.g., [67,68,40,36,79]). In particular, properties of the quaternion-valued Fourier transform find wide applications in image and signal processing (see e.g., [40]).…”

Section: The Zero Divergence Condition and Subclasses Of Meridional Ementioning

confidence: 99%

The Static Maxwell System in Three Dimensional Axially Symmetric Inhomogeneous Media and Axially Symmetric Generalization of the Cauchy–Riemann System

Bryukhov¹,

Kähler

2016

Adv. Appl. Clifford Algebras

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This paper extends our previous joint paper with Kähler, published in 2017, on problems of the static Maxwell system in three dimensional axially symmetric inhomogeneous media. Applied pseudoanalytic function theory, developed by Kravchenko, allows us to characterize new subclasses of meridional and transverse electrostatic fields in axially symmetric inhomogeneous media in two dimensional setting. Quaternionic analysis in R 3 , using analytic solutions of the system (R), allows us to characterize new subclasses of harmonic solutions of the static Maxwell system in homogeneous media in three dimensional setting. Leutwiler in 1992 initiated new approach of modified quaternionic analysis in R 3 , using analytic solutions of a hyperbolic non-Euclidean modification (H) of the system (R). Applications of modified quaternionic analysis and contemporary hyperbolic function theory in problems of geometrical optics and geo-electrostatics of inhomogeneous media are presented now. Some new classes of electrostatic potentials in three dimensional setting are implemented using Bessel functions. The singular sets of the electric field gradient tensor allow us to characterize new geometric properties of meridional fields in axially symmetric inhomogeneous media in the context of generalized axially symmetric potential theory (GASPT). Fueter construction allows us to present a wide range of meridional electrostatic models.Mathematics Subject Classification (2010). Primary 35Q61, 78A30; Secondary 35Q05, 30G20, 30C65, 30G35.Keywords. electrostatic fields; inhomogeneous isotropic media; modified quaternionic analysis; GASPT; the Fueter holomorphic potential; the Euler's Gamma function of the reduced quaternionic argument. Generalized Hyperbolic Non-Euclidean Modification of the System (R) and a Class of α-Hyperbolic Harmonic Electrostatic Potentials in Three Dimensional Setting in Cartesian CoordinatesAs seen from the system (2.3), new extensions of the system (H) require to study electrostatic models in special plane-layered inhomogeneous isotropic

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Section: The Zero Divergence Condition and Subclasses Of Meridional Ementioning

confidence: 99%

The Static Maxwell System in Three Dimensional Axially Symmetric Inhomogeneous Media and Axially Symmetric Generalization of the Cauchy–Riemann System

Bryukhov¹,

Kähler

2016

Adv. Appl. Clifford Algebras

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“…The vector derivative in R n is defined by This theorem is known as the fundamental theorem of Clifford's geometric calculus [9]. As a consequence one has Clifford-Cauchy's theorem: y − x |y − x| n n(y)f (y)dy [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and…”

Section: Monogenic Functionsmentioning

confidence: 99%

“…The image of the Clifford-Fourier transform of an integrable function is a continuous function that vanishes at infinity, i.e. if C 0 denotes the space of continuous functions defined in R n with values in G 2n that vanish at infinity, one has [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] Riemann-Lebesgue theorem.…”

Section: Operational Calculus Of F Omentioning

confidence: 99%

“…This makes the convolution theorem look rather complicated. This kernel had already been introduced as a theorical concept in Clifford analysis [6], and Sommen studied it in connection with generalizations of the Laplace, Cauchy and Hilbert transform [11]. In [8], Ebling and Scheuermann study the convolution of their Clifford-Fourier transforms: for f : R 2 → G 2…”

Section: Introductionmentioning

confidence: 99%

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The Clifford-Fourier Transform $${\mathcal {F}_o}$$ and Monogenic Extensions

López

Sánchez

2010

Adv. Appl. Clifford Algebras

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In the last decade several versions of the Fourier transform have been formulated in the framework of Clifford algebra. We present a (Clifford-Fourier) transform, constructed using the geometric properties of Clifford algebra. We show the corresponding results of operational calculus, and a connection between the Fourier transform and this new transform. We obtain a technique to construct monogenic extensions of a certain type of continuous functions, and versions of the PaleyWiener theorems are formulated.

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“…Also in Clifford analysis -a direct and elegant generalization to higher dimension of the theory of holomorphic functions in the complex plane -extensive use is made of the classical multi-dimensional Fourier transform. The idea of generalizing the Fourier Transform to the Clifford analysis setting was already performed by Sommen in [8,9] where a generalized Fourier transform was introduced in connection with similar generalizations of the Cauchy, Hilbert, and Laplace transforms; its definition is based on an exponential function which is a natural generalization of the classical Fourier kernel.…”

Section: Introductionmentioning

confidence: 99%