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.

We study discrete Hilbert boundary value problems in the case of the upper half lattice. The solutions are given in terms of the discrete Cauchy transforms for the upper and lower half space while the study of their solvability is based on the discrete Hardy decomposition for the half lattice. Furthermore, the solutions are proved to converge to those of the associated continuous Hilbert boundary value problems.

In this paper, we propose a novel (n, n) secret image sharing scheme. The construction of shares is based on matrix multiplication and the revealing is based on addition. The proposed scheme has no pixel expansion and can reconstruct the secret image precisely. It can be directly used to share grayscale images and can be easily extended to deal with the binary and color images. Experimental results show that the proposed scheme is efficient.

We consider Riemann-Hilbert boundary value problems (for short RHBVPs) with variable coefficients for axially symmetric monogenic functions defined in axial symmetric domains. This is done by constructing a method to reduce the RHBVPs for axially symmetric monogenic functions defined in four-dimensional axial symmetric domains into the RHBVPs for analytic functions defined over the complex plane. Then we derive solutions to the corresponding Schwarz problem. Finally, we generalize the results obtained to null-solutions of (D -α)φ = 0, α ∈ R, where R denotes the field of real numbers.MSC: Primary 30E25; 35Q15; 31A25; 31B20; secondary 31B10; 35J56; 35J58

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