Min Ku

Riemann-Hilbert problems for poly-Hardy space on the unit ball

In this paper, we focus on a Riemann–Hilbert boundary value problem (BVP) with a constant coefficients for the poly-Hardy space on the real unit ball in higher dimensions. We first discuss the boundary behaviour of functions in the poly-Hardy class. Then we construct the Schwarz kernel and the higher order Schwarz operator to study Riemann–Hilbert BVPs over the unit ball for the poly-Hardy class. Finally, we obtain explicit integral expressions for their solutions. As a special case, monogenic signals as elements in the Hardy space over the unit sphere will be reconstructed in the case of boundary data given in terms of functions having values in a Clifford subalgebra. Such monogenic signals represent the generalization of analytic signals as elements of the Hardy space over the unit circle of the complex plane.

Discrete Hilbert boundary value problems on half lattices

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.

Half Dirichlet problem for the Hölder continuous matrix functions in Hermitian Clifford analysis

The simultaneous null solutions of the two complex Hermitian Dirac operators are focused on Hermitian Clifford analysis, where the Hermitian Cauchy integral was constructed in the framework of circulant (2 × 2) matrix functions. Under this setting, we will present the half Dirichlet problem with boundary spaces of Hölder continuous circulant (2 × 2) matrix functions on the sphere of even-dimensional Euclidean space. We will give the unique solution to it merely by using the Hermitian Cauchy transformation, get the solutions to the Dirichlet problem on the unit ball for Hölder continuous circulant (2 × 2) matrix functions as the boundaries and the solutions to the classical Dirichlet problem as the special case, and derive a decomposition of the Poisson kernel for matrix Laplace operator.

FFT formulations of adaptive Fourier decomposition

Adaptive Fourier decomposition (AFD) has been found to be among the most effective greedy algorithms. AFD shows an outstanding performance in signal analysis and system identification. As compensation of effectiveness, the computation complexity is great, that is especially due to maximal selections of the parameters. In this paper, we explore the discretization of the 1-D AFD integration via with discrete Fourier transform (DFT), incorporating fast Fourier transform (FFT). We show that the new algorithm, called FFT-AFD, reduces the computational complexity from O(MN2) to O(MNlogN), the latter being the same as FFT. Through experiments, we verify the effectiveness, accuracy, and robustness of the proposed algorithm. The proposed FFT-based algorithm for AFD lays a foundation for its practical applications.

Riemann-Hilbert problems for null-solutions to iterated generalized Cauchy-Riemann equations in axially symmetric domains

We consider Riemann-Hilbert boundary value problems (for short RHBVPs) with variable coefficients for axially symmetric poly-monogenic functions, i.e., null-solutions to iterated generalized Cauchy-Riemann equations, defined in axially symmetric domains. This extends our recent results about RHBVPs with variable coefficients for axially symmetric monogenic functions defined in four-dimensional axially symmetric domains. First, we construct the Almansi-type decomposition theorems for poly-monogenic functions of axial type. Then, making full use of them, we give the integral representation solutions to the RHBVP considered. As a special case, we derive solutions to the corresponding Schwarz problem. Finally, we generalize the result obtained to functions of axial type which are null-solutions to perturbed iterated generalized Cauchy-Riemann equations D α k ? = 0 , k ? 2 ( k ? N ) , α ? R .

Dirichlet-Type Problems for the Iterated Dirac Operator on the Unit Ball in Clifford Analysis

We study a class of Dirichlet-type problems for null solutions to iterated Dirac operators on the unit ball of R n with boundary data given by function in Open image in new window (1<p<+∞). Applying Almansi-type decomposition theorems for null solutions to iterated Dirac operators, our Dirichlet-type problems for null solutions to iterated Dirac operators is transferred to Dirichlet-type problems for monogenic functions or harmonic functions. By introducing shifted Euler operators and making use of Clifford-Cauchy transform, we get its unique solution and its integral representation.

Solutions to a class of polynomially generalized Bers–Vekua equations using Clifford analysis

In this paper a class of polynomially generalized Vekua–type equations and of polynomially generalized Bers–Vekua equations with variable coefficients defined in a domain of Euclidean space are discussed. Using the methods of Clifford analysis, first the Fischer–type decomposition theorems for null solutions to these equations are obtained. Then we give, under some conditions, the solutions to the polynomially generalized Bers–Vekua equation with variable coefficients. Finally, we present the structure of the solutions to the inhomogeneous polynomially generalized Bers–Vekua equation.

Fast algorithm of adaptive Fourier series

Adaptive Fourier decomposition (AFD, precisely 1-D AFD or Core-AFD) was originated for the goal of positive frequency representations of signals. It achieved the goal and at the same time offered fast decompositions of signals. There then arose several types of AFDs. AFD merged with the greedy algorithm idea, and in particular, motivated the so-called pre-orthogonal greedy algorithm (Pre-OGA) that was proven to be the most efficient greedy algorithm. The cost of the advantages of the AFD type decompositions is, however, the high computational complexity due to the involvement of maximal selections of the dictionary parameters. The present paper offers one formulation of the 1-D AFD algorithm by building the FFT algorithm into it. Accordingly, the algorithm complexity is reduced, from the original

Single periodic Riemann problems for null-solutions to polynomially Cauchy–Riemann equations

\mathcal{O}(M N^2)

Riemann Boundary Value Problems on the Sphere in Clifford Analysis

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