Discrete Clifford Analysis: A Germ of Function Theory

Discrete Clifford analysis is a higher dimensional discrete function theory based on skew Weyl relations. It is centered around the study of Clifford algebra valued null solutions, called discrete monogenic functions, of a discrete Dirac operator, i.e. a first order, Clifford vector valued difference operator. In this contribution, we establish a Cauchy‐Kovalevskaya extension theorem for discrete monogenic functions defined on the grid Zhm of m‐tuples of integer multiples of a variable mesh width h. Convergence to the continuous case is investigated. As illustrative examples we explicitly construct the Cauchy‐Kovalevskaya extensions of the discrete delta function and of a discretized exponential.

Rotations in discrete Clifford analysis

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 paper we work in the `split' discrete Cli fford analysis setting, i.e. the m-dimensional function theory concerning null-functions of the discrete Dirac operator d, defi ned on the grid Zm, involving both forward and backward di fferences. This Dirac operator factorizes the (discrete) Star-Laplacian (Delta*= d2). We show how the space Hk of discrete k-homogeneous spherical harmonics, which is a reducible so(m;C)-representation, may explicitly be decomposed into 2^2m isomorphic copies of irreducible so(m;C)-representations with highest weight (k, 0, ..., 0). The key element is the introduction of 2^2m idempotents, dividing the discrete Cli fford algebra in 2^2m subalgebras of dimension (k+m-1,k) - (k+m-3, k).

We develop a linear theory of discrete complex analysis on general quad-graphs, continuing and extending previous work of Duffin, Mercat, Kenyon, Chelkak and Smirnov on discrete complex analysis on rhombic quad-graphs. Our approach based on the medial graph yields more instructive proofs of discrete analogs of several classical theorems and even new results. We provide discrete counterparts of fundamental concepts in complex analysis such as holomorphic functions, derivatives, the Laplacian, and exterior calculus. Also, we discuss discrete versions of important basic theorems such as Green's identities and Cauchy's integral formulae. For the first time, we discretize Green's first identity and Cauchy's integral formula for the derivative of a holomorphic function. In this paper, we focus on planar quad-graphs, but we would like to mention that many notions and theorems can be adapted to discrete Riemann surfaces in a straightforward way. In the case of planar parallelogram-graphs with bounded interior angles and bounded ratio of side lengths, we construct a discrete Green's function and discrete Cauchy's kernels with asymptotics comparable to the smooth case. Further restricting to the integer lattice of a two-dimensional skew coordinate system yields appropriate discrete Cauchy's integral formulae for higher order derivatives.

Clifford analysis is a higher-dimensional function theory offering a refinement of classical harmonic analysis. It is centred around monogenic functions, i.e. null solutions of the rotation invariant vector-valued Dirac operator. In this context, generalizations of classical orthogonal polynomials on the real line have been introduced in order to use them as building blocks for wavelets, as has been the case, e.g. for the generalized Clifford–Hermite polynomials. More recently, Hermitean Clifford analysis has emerged as a new branch of Clifford analysis, offering yet a refinement of the standard Euclidean case; it focuses on simultaneous null solutions of two Hermitean Dirac operators which are invariant under the action of the unitary group. In this Hermitean setting, generalized Hermitean Clifford–Hermite polynomials are constructed, starting from a Rodrigues formula involving both Hermitean Dirac operators. In this paper, we establish their mutual orthogonality relations w.r.t. a well-chosen weight function.

A growing complexity of modern engineering problems emphasises the need to methods working with discrete mathe- matical structures, such as methods of discrete potential theory and discrete function theory, are gaining more and more popularity nowadays. However, solution procedure of boundary value problems by these methods requires at first a construction of the dis- cretisation of a given domain. Thus, in this paper, we present a general discretisation algorithm for discrete potential and function theories. The algorithm allows a consistent discretisation of a domain for interior and exterior boundary value problems based only on a definition of a given continuous domain; and the complete discretisation procedure is based on the application of well-known set-theoretic operations originating from the characterisation of mesh points by help of star-Laplacian. Thus, the algorithm results in a rectangular lattice established over an arbitrary two-dimensional domain and its exterior.

This article focuses on the integral theory of discrete regular quaternionic functions and the connection between discrete and classical quaternionic analysis. We develop a new approach to the discrete integral theory based on the concepts ‘discrete boundary measure’ and ‘discrete normal vector’. By combining the discrete integral theory and our results on discrete approximations to domains in \({\mathbb {R}}^4\), we reveal a convergence relation between the discrete and continuous regular functions: a quaternionic function is regular if and only if it is the scaling limit of some discrete regular functions.

We establish dynamical localization for random Dirac operators on the d-dimensional lattice, with \(d\in \left\{ 1, 2, 3\right\} \), in the three usual regimes: large disorder, band edge and 1D. These operators are discrete versions of the continuous Dirac operators and consist in the sum of a discrete free Dirac operator with a random potential. The potential is a diagonal matrix formed by different scalar potentials, which are sequences of independent and identically distributed random variables according to an absolutely continuous probability measure with bounded density and of compact support. We prove the exponential decay of fractional moments of the Green function for such models in each of the above regimes, i.e., (j) throughout the spectrum at larger disorder, (jj) for energies near the band edges at arbitrary disorder and (jjj) in dimension one, for all energies in the spectrum and arbitrary disorder. Dynamical localization in theses regimes follows from the fractional moments method. The result in the one-dimensional regime contrast with one that was previously obtained for 1D Dirac model with Bernoulli potential.

We define a discrete Laplace-Beltrami operator for simplicial surfaces. It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finite-elements Laplacian (the so called ``cotan formula'') except that it is based on the intrinsic Delaunay triangulation of the simplicial surface. This leads to new definitions of discrete harmonic functions, discrete mean curvature, and discrete minimal surfaces. The definition of the discrete Laplace-Beltrami operator depends on the existence and uniqueness of Delaunay tessellations in piecewise flat surfaces. While the existence is known, we prove the uniqueness. Using Rippa's Theorem we show that, as claimed, Musin's harmonic index provides an optimality criterion for Delaunay triangulations, and this can be used to prove that the edge flipping algorithm terminates also in the setting of piecewise flat surfaces.

Some properties of hypercomplex functions (the null solutions of the polynomial Dirac operators in Rn+1) in Clifford Analysis are discussed, their hypercomplex functional calculi for an n-tuple non-commuting self-adjoint operators A are constructed by the use of Cauchy integral formulas, the polynomial approaches to functional calculi are also considered. Although these hypercomplex function theories have different representative forms, their hypercomplex functional calculi are the same as the monogenic functional calculus.

Discrete function theory is a natural extension of the continuous theory to functions defined on lattices. The idea of the discrete function theory is to work directly with discretised domains (lattices) and to transfer all important properties from the continuous case to the discrete level. In the field of boundary value problems it is more beneficial to work with rectangular lattices, i.e. allowing two different stepsizes. Thus, the aim of this paper is to present the extension of the discrete function theory to rectangular lattices. Particularly, we present the discrete analogue of the Borel–Pompeiu formula. Analogously to the continuous theory it is the core of the developed discrete function theory and the related operator calculus.

This paper investigates the effects of bulk acoustic wave (BAW) radiation on the admittance of interdigital transducers (IDTs) with significant internal reflections of shear horizontal type (SH-type) leaky surface acoustic waves (SAWs) on 36/spl deg/YX-LiTaO/sub 3/ (36-LT). Theoretical analysis is made by using the discrete Green function theory, and synchronous one-port resonators are analyzed. It is shown that the BAW radiation significantly affects the IDT characteristics even for resonators; under certain circumstances, BAWs launched from an IDT are converted into SH-type SAWs by the strong internal reflections, and they interact with the BAWs radiated by the IDT. Then the net amount of the radiated BAW power is highly dependent upon the number of IDT finger pairs. For the precise simulation of devices based on the SH-type SAWs with strong internal reflections, the BAW radiation should carefully be taken into consideration.

Blotekjaer's theory, extended by Hashimoto and coworkers, for analyzing surface transverse waves has been modified in the part concerning the effect of finite electrode thickness. First order Datta-Hunsinger's surface boundary conditions have been introduced in the effective permittivity of the substrate, to account for the mechanical loading. The dispersion STW velocity curves have been obtained over the entire frequency stopband, including the interval where intensive excitation of bulk modes occurs. The effective permittivity also allows obtaining the electromechanical coupling of the grating mode. Its numerical analysis reveals a strong dependence on the metalization ratio and mass-loading, in close agreement with results from Hasimoto's discrete Green function theory.

Special functions in Clifford analysis and axial symmetry