Harmonic Analysis and Hypercomplex Function Theory in Co-dimension One

Abstract

Fundamentals of a function theory in co-dimension one for Clifford algebra valued functions over R n+1 are considered. Special attention is given to their origins in analytic properties of holomorphic functions of one and, by some duality reasons, also of several complex variables. Due to algebraic peculiarities caused by non-commutativity of the Clifford product, generalized holomorphic functions are characterized by two different but equivalent properties: on one side by local derivability (existence of a well defined derivative related to co-dimension one) and on the other side by differentiability (existence of a local approximation by linear mappings related to dimension one). As important applications sequences of harmonic Appell polynomials are considered whose definition and explicit analytic representations rely essentially on both dual approaches.

A Sturm‐Liouville equation on the crossroads of continuous and discrete hypercomplex analysis

“…and obtain the same as in the classical case (41) Theorem 1. A sequence of polynomials  in ℙ is an Appell sequence if and only if…”

“…Proofs and further details, including series or corresponding alternating differential forms, can be found in Malonek et al. 40,41,47 (i) If = ( 1 , … , ) is a multi-index, all homogeneous monogenic polynomials of degree | | = can be obtained as linear combinations (from the left or from the right)…”

“…15,40 A short overview can also be found in a very recent survey. 41 The simplest example of complexification for working with the algebra of complex numbers in the sense mentioned by V. I. Arnold is the identification of R 2 with C, formally expressed by the two conjugate variables z = x + iy and z = x − i𝑦. For a system of two real differentiable functions u = u(x, y) and v = v(x, y) and demanding that 𝑓 = 𝑓 (z, z) is complex differentiable with respect to z, this leads to the usual system of Cauchy-Riemann equations.…”

“…25 Using a differential form calculus 2 and a corresponding Cauchy integral, this paper contains the proof that the existence of the hypercomplex derivative is necessary and sufficient ¶ for defining monogenic functions, that is, solutions of the generalized Cauchy-Riemann system. 41 The necessary topological basics are almost obvious.…”

Non-symmetric Number Triangles Arising from Hypercomplex Function Theory in $$\mathbb {R}^{n+1}$$

“…The case n > 1 extends the complex case to paravector valued functions of n hypercomplex non-commutative variables or, by duality, of one paravector valued variable. For more details we refer to the articles [6,8,9,11].…”