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This paper aims at proving the boundedness property of multidimensional singular integral operators associated with (φ , ψ) -harmonic functions, which are connected by the use of two orthogonal basis of the Euclidean space R m . Besides, necessary and sufficient conditions for the solvability of the ∂ ¯ -problem for such (φ , ψ) -harmonic functions are described. The basic devices that make it possible to state and prove both results are borrowed from Clifford analysis. [ABSTRACT FROM AUTHOR]

Classical Segal–Bargmann theory studies three Hilbert space unitary isomorphisms that describe the wave-particle duality and the configuration space-phase space. In this work, we generalized these concepts to Clifford algebra-valued functions. We establish the unitary isomorphisms among the space of Clifford algebra-valued square-integrable functions on R n with respect to a Gaussian measure, the space of monogenic square-integrable functions on R n + 1 with respect to another Gaussian measure and the space of Clifford algebra-valued linear functionals on symmetric tensor elements of R n . [ABSTRACT FROM AUTHOR]

In this paper we study a generalization of Clifford algebra depending on parameters introduced by Tutschke and Vanegas in 2008. We also introduce some related notions, a Cauchy–Pompeiu integral formula, and two boundary value problems for monogenic functions with values in this Cliford algebra. [ABSTRACT FROM AUTHOR]

In this paper, based on a proposed notion of generalized conjugate harmonic pairs in the framework of complex Clifford analysis, necessary and sufficient conditions for the solvability of inhomogeneous perturbed generalized Moisil–Teodorescu systems in higher dimensional Euclidean spaces are proved. As an application, we derive corresponding solvability conditions for the inhomogeneous Maxwell's equations. [ABSTRACT FROM AUTHOR]

In this paper, we mainly consider the Riemann boundary value problems for lower dimensional non-commutative Clifford algebras valued monogenic functions. The solutions are given in an explicit way and concrete examples are presented to illustrate the results. [ABSTRACT FROM AUTHOR]

In this paper, we present the algebraic properties of dual quaternions and define a slice regularity of a dual quaternionic function. Since the product of dual quaternions is non-commutative, slice regularity is derived in two ways. Thereafter, we propose the Cauchy-Riemann equations and a power series corresponding to dual quaternions. [ABSTRACT FROM AUTHOR]

The theory of polynomial bases in one complex variable, as given by Whittaker and Cannon, was extended partially to the scope of Clifford analysis, where many results have been handled from different aspects. The emphasis of this paper is on introducing a certain type of bases of axially monogenic polynomials generated by the maximum modulus base. Determining the convergence properties of such bases is closely related to the growth behavior of the associated entire axially monogenic functions. We start by investigating the presented bases of monogenic polynomials associated with entire axially monogenic functions in the way we are going to indicate. Then, we show how this leads us to certain relations involving maximum moduli of an entire axially monogenic function on a sequence of balls with increasing radii. In this concern, we point out that Hadamard's three hyper-balls theorem can be employed to justify some of our results. The significance of this study is due to the variety of the results, which are not normally expected. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavor of the real Clifford Fourier transform. We introduce the Clifford Fourier transform for measure. Moreover, we give some of its properties. Analogue of Bochner Minlos theorem is established in real Clifford algebra. [ABSTRACT FROM AUTHOR]

This paper deals with generalized axially symmetric potentials (GASP) which are solutions of a Weinstein-type equation in R 3 ∂ 2 Φ ∂ x 2 + ∂ 2 Φ ∂ y 2 + ∂ 2 Φ ∂ z 2 + 2 (m + 1) z ∂ Φ ∂ z = 0 , m ∈ N. GASP have been investigated by several generations of mathematicians. In this paper, we introduce an explicit representation of the GASP by differential operators via harmonic functions in Clifford analysis setting. [ABSTRACT FROM AUTHOR]

This paper deals with the static Maxwell system d i v (Φ E →) = 0 , c u r l E → = 0 , (x 0 , x 1 , x 2) ∈ R 3. The system is reformulated in quaternion analysis by Kravchenko in the form L F = 0 with L F = D F + F α . We consider special cases of the coefficient function Φ = Φ 0 (x 0) Φ 1 (x 1) Φ 2 (x 2) and prove the existence of four generalized Cauchy kernels of the operator L . We construct four explicit generalized Cauchy kernels in the case Φ = x 0 2 p x 1 2 m x 2 2 n . [ABSTRACT FROM AUTHOR]

Recently 3D image processing has interested researchers in both theoretical and applied fields and thus has constituted a challenging subject. Theoretically, this needs suitable functional bases that are easy to implement by the next. It holds that Clifford wavelets are main tools to achieve this necessity. In the present paper we intend to develop some new classes of Clifford wavelet functions. Some classes of new monogenic polynomials are developed firstly from monogenic extensions of 2-parameters Clifford weights. Such classes englobe the well known Jacobi, Gegenbauer and Hermite ones. The constructed polynomials are next applied to develop new Clifford wavelets. Reconstruction and Fourier-Plancherel formulae have been proved. Finally, computational examples are developed provided with graphical illustrations of the Clifford mother wavelets in some cases. Some graphical illustrations of the constructed wavelets have been provided and finally concrete applications in biofields have been developed dealing with EEG/ECG and Brain image processing. [ABSTRACT FROM AUTHOR]

The paper provides integral representations for solutions to a certain first order partial differential equation natural arising in the factorization of the Lamé—Navier system with the help of Clifford analysis techniques. These representations look like in spirit to the Borel—Pompeiu and Cauchy integral formulas both in three and higher dimensional setting. [ABSTRACT FROM AUTHOR]

In this paper, we study the Bargmann–Radon transform and a class of monogenic functions called axially monogenic functions. First, we compute the explicit formula of the Bargmann–Radon transform for axially monogenic functions, by making use of the Funk–Hecke theorem. Then we present the explicit form of the general Cauchy–Kowalewski extension for radial function. Finally, by making use of the results we obtained, we give an application of the Bargmann–Radon transform for Cauchy–Kowalewski extension. [ABSTRACT FROM AUTHOR]

Abstract This article mainly studies the boundary value problems for hypergenic function vectors in Clifford analysis. Firstly, some properties of hypergenic quasi-Cauchy type integrals are discussed. Then, by the Schauder fixed point theorem the existence of the solution to the nonlinear boundary value problem is proved. Finally, using the compression mapping principle the existence and uniqueness of the solution to the linear boundary value problem are proved.

In recent work a radial deformation of the Fourier transform in the setting of Clifford analysis was introduced. The key idea behind this deformation is a family of new realizations of the Lie superalgebra osp (1 | 2) in terms of a so-called radially deformed Dirac operator D depending on a deformation parameter c such that for c = 0 the classical Dirac operator is reobtained. In this paper, several versions of the Paley–Wiener theorems for this radially deformed Fourier transform are investigated, which characterize the supports of functions associated to this generalized Fourier transform in Clifford analysis. [ABSTRACT FROM AUTHOR]

We investigate some properties of solutions to Dirac operators with gradient potentials. Solutions to Dirac operators with gradient potentials are called monogenic functions with respect to the potential functions. We establish a Borel–Pompeiu formula, and obtain Cauchy integral formula and mean value formula about such functions. Based on integral formulas, we prove some geometric properties of monogenic functions with respect to the potential functions and construct some integral transforms. The boundedness of integral transforms in Hölder space is given. We prove the Plemelj–Sokhotski formula and the Painlevé theorem. As applications, we firstly prove that a kind of Riemann–Hilbert problem for monogenic functions with respect to the potential functions is solvable. Explicit representation formula of the solution is also given. We then establish solvability conditions of a kind of general Riemann–Hilbert problem for monogenic functions with respect to the potential functions. Finally, we give some examples about the above Riemann–Hilbert problem. [ABSTRACT FROM AUTHOR]

In this paper we derive a Cauchy integral representation formula for the solutions of the iterated sandwich equation ∂ x ̲ 2 k - 1 f ∂ x ̲ = 0 , where k is a positive integer and ∂ x ̲ stands for the Dirac operator in the Euclidean space R m . We call these solutions (2 k - 1) -infrapolymonogenic functions (or simply infrapolymonogenic if no confusion can arise). For k = 1 the derived formula becomes the Cauchy integral representation formula recently obtained in Moreno-García et al. (Adv Appl Clifford Algebras 27(2):1147–1159, 2017) for inframonogenic functions. [ABSTRACT FROM AUTHOR]

In this paper, we consider functions satisfying the sandwich equation ∂x_f∂x_=0, where ∂x_ stands for the Dirac operator in Rm. Such functions are referred as inframonogenic and represent an extension of the monogenic functions, ie, null solutions of ∂x_. In particular, for odd m, we prove that a C2‐function is both inframonogenic and harmonic in Ω⊂Rm if and only if it can be represented in Ω as f=f1+f2+f3x_+x_f4,where f1 and f2 are, respectively, left and right monogenic functions in Ω, while f3 and f4 are two‐sided monogenic functions there. Finally, in deriving some applications of our results, we have made use of the deep connection between the class of inframonogenic vector fields and the universal solutions of the Lamé‐Navier system in R3. [ABSTRACT FROM AUTHOR]

The notion of the generalized Hadamard product of bases in Clifford analysis is introduced. Certain convergence properties of such generalized Hadamard product bases in terms of those of their constituent bases are established. Unlike the complex case, some of these convergence properties may not be interchanged. [ABSTRACT FROM AUTHOR]

In this paper, we prove several versions of the real Paley–Wiener theorems for a fractional Clifford–Fourier transform which depends on two numerical parameters and paves the way in some sense for a functional calculus approach to generalizing the Fourier transform to Clifford analysis. [ABSTRACT FROM AUTHOR]

This paper deals with a type of generalized monogenic functions in Clifford analysis, the so-called Weinstein k-monogenic function. We will give an explicit representation of the Weinstein k-monogenic functions by differential operators of Bauer’s type in the cases k = 2 m , m ∈ N ∗ . [ABSTRACT FROM AUTHOR]

Nowadays, 3D images processing constitutes a challenging topic in many scientific fields such as medicine, computational physics, informatics. Therefore, the construction of suitable functional bases that allow computational aspects to be easily done is a necessity. Wavelets and Clifford algebras are ones of the most import mathematical tools for achieving such necessities. In the present work, new classes of wavelet functions are presented in the framework of Clifford analysis. Firstly, some classes of new monogenic polynomials are provided based on two-parameter weight functions. Such classes englobe the well-known Gegenbauer and Hermite ones. The discovered polynomial sets are next applied to introduce new wavelet functions. Reconstruction formula and Fourier–Plancherel rules have been proved. Computational concrete examples are developed by means of some illustrative examples with graphical representations of the Clifford mother wavelets in some cases. These discovered wavelets have been applied by the next for modeling some biomedical signals such as EEG, ECG and proteins. [ABSTRACT FROM AUTHOR]

In the framework of Clifford analysis, we consider the Paley‐Wiener type theorems for a generalized Clifford‐Fourier transform. This Clifford‐Fourier transform is given by a similar operator exponential as the classical Fourier transform but containing generators of Lie superalgebra. [ABSTRACT FROM AUTHOR]

Using the method of normalized systems of functions, we study one representation of real analytic functions by monogenic functions (i.e., solutions of Dirac equations), which is an Almansi's formula of infinite order. As applications of the representation, we construct solutions of the inhomogeneous Dirac and poly-Dirac equations in Clifford analysis. [ABSTRACT FROM AUTHOR]

Transmission boundary value problems for the time-harmonic Maxwell system, the Helmholtz operator, and the perturbed Dirac operator are formulated, using Clifford algebras, on bounded Lipschitz domains of R m with m ≥ 3. It is shown how the Dirac problem decouples into several Maxwell and Helmholtz problems. Necessary and sufficient conditions are provided for well-posedness in each case, and for the Dirac problem to be equivalent to one or several independent Maxwell problems. [ABSTRACT FROM AUTHOR]

B-splines B q of quaternionic order q, for short quaternionic B-splines, are quaternion-valued piecewise Müntz polynomials whose scalar parts interpolate the classical Schoenberg splines B n , n ∈ N , with respect to degree and smoothness. They in general do not satisfy the interpolation property B q (n) = δ n , 0 , n ∈ Z . However, the application of the interpolation filter (∑ k ∈ Z B q ^ (ξ + 2 π k)) - 1 —if well-defined—in the frequency domain yields a cardinal fundamental spline of quaternionic order that satisfies the interpolation property. We handle the ambiguity of the quaternion-valued exponential function appearing in the denominator of the interpolation filter and relate the filter to interesting properties of a quaternionic Hurwitz zeta function and the existence of complex quaternionic inverses. Finally, we show that the cardinal fundamental splines of quaternionic order fit into the setting of Kramer's Lemma and allow for a family of sampling, respectively, interpolation series. [ABSTRACT FROM AUTHOR]

In this article, we study the steady generalized Navier-Stokes equations in a half-space in the setting of variable exponent spaces. We first establish variable exponent spaces of Clifford-valued functions in a half-space. Then, using this operator theory together with the contraction mapping principle, we obtain the existence and uniqueness of solutions to the stationary Navier-Stokes equations and Navier-Stokes equations with heat conduction in a half-space under suitable hypotheses.

In this paper, we give the notation and properties of the vectorial form of biquaternions. The differential operators and calculations result from a modified multiplication with the vectorial form. [ABSTRACT FROM AUTHOR]

In recent years, much attention has been paid to the role of classical special functions of a real or complex variable in mathematical physics, especially in boundary value problems (BVPs). In the present paper, we propose a higher‐dimensional analogue of the generalized Bessel polynomials within Clifford analysis via a special set of monogenic polynomials. We give the definition and derive a number of important properties of the generalized monogenic Bessel polynomials (GMBPs), which are defined by a generating exponential function and are shown to satisfy an analogue of Rodrigues' formula. As a consequence, we establish an expansion of particular monogenic functions in terms of GMBPs and show that the underlying basic series is everywhere effective. We further prove a second‐order homogeneous differential equation for these polynomials. [ABSTRACT FROM AUTHOR]

Abstract First, a kind of generalized Teodorescu operator which is related to the k-regular functions are defined. Then the boundedness and the Hölder continuity of the generalized Teodorescu operator are discussed. Finally, the stability and error estimate of the generalized Teodorescu operator are given when the boundary surface of the integral domain Ω is perturbed.