Basic sets of pofynomials in clifford analysis

In a recent paper [Trans. Amer. Math. Soc. 378 (2025), pp. 851–883], the concept of generalized partial-slice monogenic (or regular) function was introduced over Clifford algebras. The present paper shall extend the study of generalized partial-slice monogenic functions from the associative case of Clifford algebras to non-associative alternative algebras, such as octonions. The new class of functions encompasses the regular functions [Rend. Sem. Mat. Univ. Padova 50 (1973), pp. 251–267] and slice regular functions [Rocky Mountain J. Math. 40 (2010), no. 1, 225–241] over octonions, indeed both appear in the theory as special cases. In the non-associative setting of octonions, we shall develop some fundamental properties such as identity theorem, Representation Formula, Cauchy (and Cauchy-Pompeiu) integral formula, maximum modulus principle, Fueter polynomials, Taylor series expansion. As a complement, the paper also introduces and discusses the notion of generalized partial-slice (and regular) functions. Although the study is limited to the case of octonions, it is clear from the statements and the arguments in the proofs that the results hold more in general in real alternative algebras equipped with a notion of conjugation.

ON RIEMANN-HILBERT PROBLEM FOR REGULAR FUNCTIONS WITH VALUES IN A CLIFFORD ALGEBRA

From classical complex analysis it is well-known that the index of the Riemann-Hilbert problem is connected with the winding number. How to generalize the winding number to a rotational number or rotation is explained in [nf] and also in [krza]. How to get a connection between the index and the rotation can be found in [dei]. For matrices of singular integral operators a rotation can be defined which is connected with the index of this system of singular integral operators. Because Clifford analysis is a higher dimensional analog to complex analysis there should be some analogous behavior of singular integrals. In fact, we will show that the index of singular integral operators with paravector-valued symbols in Clifford algebras can be connected with the so-called wrapping number which is the higher dimensional analog to the winding number. The proof is rather elementary.

In the classical complex analysis of several variables Lars Hoermander had been proven for Hartog's extension theorem. His proof is based on solving the inhomogeneous Cauchy–Riemann system. However, it is difficult when generalizing this diagram to prove this problem in Clifford analysis because of the non-commutative property. In this paper we will propose a method to overcome this difficultly and we will prove the Hartog's extension theorem for multi-regular function taking valued in Clifford algebra

Regular cost functions have been introduced recently as an extension to the notion of regular languages with counting capabilities. The specificity of cost functions is that exact values are not considered, but only estimated. In this paper, we study the strict subclass of regular temporal cost functions. In such cost functions, it is only allowed to count the number of occurrences of consecutive events. For this reason, this model intends to measure the length of intervals, i.e., a discrete notion of time. We provide various equivalent representations for functions in this class, using automata, and 'clock based' reduction to regular languages. We show that the conversions are much simpler to obtain, and much more efficient than in the general case of regular cost functions. Our second aim in this paper is to use temporal cost function as a test-case for exploring the algebraic nature of regular cost functions. Following the seminal ideas of Schutzenberger, this results in a decidable algebraic characterization of regular temporal cost functions inside the class of regular cost functions.

This survey is intended as an overview of discrete Clifford analysis and its current developments. Since in the discrete case one has to replace the partial derivative with two difference operators, backward and forward partial difference, one needs to modify the main tools for a development of a discrete function theory, such as the replacement of a real Clifford algebra by a complexified Clifford algebra or of the classic Weyl relations by so-called S-Weyl relations. The main results, like Cauchy integral formula, Fischer decomposition, CK-extension, and Taylor series, will be derived. To give a better idea of the differences between the discrete and continuous case, this chapter contains the problem of discrete Hardy spaces as well as some discrete objects which do not have an equivalent object in continuous Clifford analysis, such as the CK-extension of a discrete Delta function.

In this article, we mainly deal with the boundary value problem for harmonic function with values in Clifford algebra: urn:x-wiley:dummy:mana201100302:equation:mana201100302-math-0001where is a Liapunov surface in , the Dirac operator , are unknown functions with values in a universal Clifford algebra Under some assumptions, we show that the boundary value problem is solvable.

For the simplicity we’ll consider here the algebra of quaternions ℍ instead of a Clifford algebra. At the same time most results hold for Clifford algebra valued functions also.

We define a new modified basis which is an association of two bases, e 1 and 2 . We give an expression of the form = 0 + 0 , where 0 is a real number and 0 is a complex number on three-dimensional real skew field. And we research the properties of regular functions with values in ternary field and reduced quaternions by Clifford analysis.

In this paper we derive for the even dimensional case a closed form of the Fourier–Borel kernel in the Clifford analysis setting. This kernel is obtained as the monogenic component in the Fischer decomposition of the exponential function \({e^{\langle \underline{x}, \underline{u} \rangle}}\) where \({\langle . , . \rangle}\) denotes the standard inner product on the m-dimensional Euclidean space. A first approach based on Clifford analysis techniques leads to a conceptual formula containing the Gamma operator and the so-called Clifford–Bessel function, two fundamental objects in the theory of Clifford analysis. To obtain an explicit expression for the Fourier–Borel kernel in terms of a finite sum of Bessel functions, this formula remains however hard to work with. To that end we have also elaborated a more direct approach based on special functions leading to recurrence formulas for a closed form of the Fourier–Borel kernel.

In this paper, we discuss the so‐called Witt basis in a Clifford algebra and we axiomatically define an algebra of abstract Hermitian vector variables similar to the ‘radial algebra’. In this setting, we introduce some linear partial differential operators and we study their resolutions. Copyright © 2002 John Wiley & Sons, Ltd.

We consider the real vector space structure of the algebra of linear endomorphisms of a finite-dimensional real Clifford algebra (2, 4, 5, 6, 7, 8). A basis of that space is constructed in terms of the operators M eI, eJ defined by x↦e I · x · e J , where the e I are the generators of the Clifford algebra and I is a multi-index (3, 7). In particular, it is shown that the family (M eI, eJ ) is exactly a basis in the even case.

In this article, we bring a new set of Bernoulli and Euler special monogenic polynomials. It is based on the umbral calculus adapted to the Clifford setting, which is equivalent to a polynomial representation using a special set of Appell polynomials. Also, we study the convergence properties (the effectiveness) for the Bernoulli and Euler polynomials in Clifford analysis. Finally, some results concerning the mode of increase (the order and type) for these sets are obtained.

The monogenic functional calculus is a means of constructing functions of a finite system of bounded or unbounded operators. For bounded noncommuting operators, a polynomial function produces a polynomial of the operators in which all possible operator orderings are equally weighted. For example, for two bounded selfadjoint operators A 1, A 2, the operator \(\frac{1} {2}(A_{1}A_{2} + A_{2}A_{1})\) is associated with polynomial \(z\mapsto z_{1}z_{2}\) in two variables by the monogenic functional calculus. The same formula applies just when the spectrum \(\sigma (\xi _{1}A_{1} +\xi _{2}A_{2})\) of a finite linear combination \(\xi _{1}A_{1} +\xi _{2}A_{2}\) of A 1 and A 2 is a subset of the real numbers for any \(\xi _{1},\xi _{2} \in \mathbb{R}\). The article begins with a discussion of Clifford algebras and Clifford analysis and points out the connection with Weyl’s functional calculus for a finite system of selfadjoint operators in Hilbert space. The construction of the Cauchy kernel in the monogenic functional calculus is achieved with the plane wave decomposition of the Cauchy kernel in Clifford analysis. The formulation applies to any finite system \((A_{1},\ldots ,A_{n})\) of closed unbounded operators such that the spectrum \(\sigma \left (\xi _{1}A_{1} +\ldots +\xi _{n}A_{n}\right )\) of the closed operator \(\xi _{1}A_{1} +\ldots +\xi _{n}A_{n}\) is contained in a two-sided sector in \(\mathbb{C}\) for almost all \(\xi \in \mathbb{R}^{n}\). The connection with harmonic analysis and irregular boundary value problems is emphasized. The monogenic functional calculus may be applied to the finite commuting system \(D_{\Sigma } = i(D_{1},\ldots ,D_{n})\) of differentiating operators on a Lipschitz surface \(\Sigma \) so that an H ∞-functional calculus \(f\mapsto f(D_{\Sigma })\) is obtained for functions f uniformly bounded and left monogenic in a sector containing almost all tangent planes to \(\Sigma \).KeywordsBounded Linear OperatorClifford AlgebraFunctional CalculusSelfadjoint OperatorMonogenic FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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 $${\mathfrak {osp}}(1|2)$$ in terms of a so-called radially deformed Dirac operator $${\mathbf {D}}$$ 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.