Harmonic Polynomials Via Differentiation

It is well-known that if p is a homogeneous polynomial of degree k in n variables, p ∈ P;, then the ordinary derivative p（）(r;) has the form A;Y（x）r;where A;is a constant and where Y is a harmonic homogeneous polynomial of degree k, Y ∈ H;, actually the projection of p onto H;. Here we study the distributional derivative p（）(r;) and show that the ordinary part is still a multiple of Y, but that the delta part is independent of Y, that is, it depends only on p-Y. We also show that the exponent 2-n is special in the sense that the corresponding results for p（）(r;)do not hold if α≠2-n. Furthermore, we establish that harmonic polynomials appear as multiples of r;when p（） is applied to harmonic multipoles of the form Y’（x）r;for some Y ∈H;.

Invertibility of Bergman Toeplitz operators with harmonic polynomial symbols

Let p be an analytic polynomial on the unit disk.We obtain a necessary and sufficient condition for Toeplitz operators with the symbol z+p to be invertible on the Bergman space when all coefficients of p are real numbers.Furthermore,we establish several necessary and sufficient,easy-to-check conditions for Toeplitz operators with the symbol z+p to be invertible on the Bergman space when some coefficients of p are complex numbers.

Rotational invariants constructed by the products of three spherical harmonic polynomials

The rotational invariants constructed by the products of three spherical harmonic polynomials are expressed generally as homogeneous polynomials with respect to the three coordinate vectors in the compact form, where the coefficients are calculated explicitly in this paper.

POLYNOMIAL SOLUTIONS TO PIEZOELECTRIC BEAMS (Ⅱ) ——ANALYTICAL SOLUTIONS TO TYPICAL PROBLEMS

For the orthotropic piezoelectric plane problem, a series of piezoelectric beams is solved and the corresponding analytical solutions are obtained with the trialand-error method on the basis of the general solution in the case of three distinct eigenvalues, in which all displacements, electrical potential, stresses and electrical displacements are expressed by three displacement functions in terms of harmonic polynomials. These problems are cantilever beam with cross force and point charge at free end, cantilever beam and simply-supported beam subjected to uniform loads on the upper and lower surfaces, and cantilever beam subjected to linear electrical potential.

POLYNOMIAL SOLUTIONS TO PIEZOELECTRIC BEAMS (Ⅰ) ——SEVERAL EXACT SOLUTIONS

For the orthotropic piezoelectric plane problem, a series of piezoelectric beams is solved and the corresponding exact solutions are obtained with the trial-anderror method on the basis of the general solution in the case of three distinct eigenvalues, in which all displacements, electrical potential, stresses and electrical displacements are expressed by three displacement functions in terms of harmonic polynomials. These problems are rectangular beams having rigid body displacements and identical electrical potential, rectangular beams under uniform tension and electric displacement as well as pure shearing and pure bending, beams of two free ends subjected to uniform electrical potential on the upper and lower surfaces.

A Weak Galerkin Harmonic Finite Element Method for Laplace Equation

In this article,a weak Galerkin finite element method for the Laplace equation using the harmonic polynomial space is proposed and analyzed.The idea of using the P_(k)-harmonic polynomial space instead of the full polynomial space P_(k)is to use a much smaller number of basis functions to achieve the same accuracy when k≥2.The optimal rate of convergence is derived in both H^(1)and L^(2)norms.Numerical experiments have been conducted to verify the theoretical error estimates.In addition,numerical comparisons of using the P_(2)-harmonic polynomial space and using the standard P_(2)polynomial space are presented.

ON APPROXIMATION BY SPHERICAL REPRODUCING KERNEL HILBERT SPACES

The spherical approximation between two nested reproducing kernels Hilbert spaces generated from different smooth kernels is investigated. It is shown that the functions of a space can be approximated by that of the subspace with better smoothness. Furthermore, the upper bound of approximation error is given.