Discrete Hardy Spaces

Abstract

We study the boundary behavior of discrete monogenic functions, i.e. null-solutions of a discrete Dirac operator, in the upper and lower half space. Calculating the Fourier symbol of the boundary operator we construct the corresponding discrete Hilbert transforms, the projection operators arising from them, and discuss the notion of discrete Hardy spaces. Hereby, we focus on the 3D-case with the generalization to the n-dimensional case being straightforward.

Relativistic Wave Equations on the Lattice: An Operational Perspective

“…In the shed of the fractional calculus formulation proposed recently by Bernstein (cf. [3,4]), it may also provides us a meaningful way to generalize the results of [5,6], where only the properties of the discrete Fourier transform depicted in [22, subsection 5.2] were taken into account.…”

“…There are already some recent contributions that recognizes that the theory of finite difference potentials presented by Gürlebeck and Sprössig on their book may also be used to describe the solution of boundary value problems on half-lattices [5,6] by a discrete version of the Hilbert transform.…”

Solutions for the Klein–Gordon and Dirac Equations on the Lattice Based on Chebyshev Polynomials

“…Interestingly enough (but not yet fully adopted or known in the community) combination of tools from finite difference potentials (cf. [19,5]) and interpolation theory (cf. [18,6]) arising in this context may also be useful in the modelling of problems of quantum field theory over the phase space hZ n × − π h , π h n (see, for instance, [23,12]).…”

“…on f (x) given by On the flavor of finite difference potentials (cf. [19,5]) such kind of quantization on the lattice that results into the aforementioned energy-momentum relation was already considered, in a hidden way, when the authors obtained integral representations involving Green's-type functions based on the fact that the restriction of the continuous Fourier transform to Q h = − π h , π h n gives an inverse for the discrete Fourier transform over hZ n (see also [24,Section 1.5], [12,Section II. ] and [16,Subsection 4.2] for further analogies and comparisons).…”

Modern Trends in Hypercomplex Analysis

“…While lately one can observe several approaches to create a discrete function theory in higher dimensions based on lattice discretizations of the Dirac operator (see [32,26,21,18,19,4,8]) they are closer in spirit to finite difference methods than finite element methods ( [5,22,23,2,6]). Nevertheless, these approaches lead to a well established function theory [17,9,10,11,12,20,7]. For a function theory in connection with the above mentioned finite element exterior calculus we do not want to be restricted to meshes coming from simplicial complexes.…”