Abstract:
Since the reform and opening up, FDI has made a significant contribution to China’s economic development; however, the crowding-out phenomenon appears inevitably in some districts and industries where FDI enters intensively. Confronted with the new environment of foreign capital use after the international financial crisis, our country has begun to adjust investment attraction work from the policy level and needs to specify the work the next step according to every region’s situation. Based on this background, this paper examined how foreign investment impact domestic investment over the 30 years, especially contrasted regional differences of crowding-in or crowding-out effect of FDI on domestic investment in eastern, central and western China, then made further analysis of the causes, in order to supply the policy makers and investors with effective references.

Abstract:
This article reviewed the researches on tax planning published in the domestic and foreign authoritative magazines, and found that scholars put their emphasis on three points: research on the motivation of tax planning; research on the measurement of tax planning; research on the economic consequences of tax planning. With the improvement of the market economy system and the tax system in our country, tax planning can help enterprises to achieve maximum direct economic benefits and value. Besides, with the four-year reform of “business tax to value-added tax” coming to the end, it is particularly important for the enterprises to establish effective corporate tax planning strategy to achieve the scientific and sustainable development.

Abstract:
The relation between Radon transform and orthogonal expansions of a function on the unit ball in $\RR^d$ is exploited. A compact formula for the partial sums of the expansion is given in terms of the Radon transform, which leads to algorithms for image reconstruction from Radon data. The relation between orthogonal expansion and the singular value decomposition of the Radon transform is also exploited.

Abstract:
The direct and inverse theorems are established for the best approximation in the weighted $L^p$ space on the unit sphere of $\RR^{d+1}$, in which the weight functions are invariant under finite reflection groups. The theorems are stated using a modulus of smoothness of higher order, which is proved to be equivalent to a $K$-functional defined using the power of the spherical $h$-Laplacian. Furthermore, similar results are also established for weighted approximation on the unit ball and on the simplex of $\RR^d$.

Abstract:
A polynomial of the form $x^\alpha - p(x)$, where the degree of $p$ is less than the total degree of $x^\alpha$, is said to be least deviation from zero if it has the smallest uniform norm among all such polynomials. We study polynomials of least deviation from zero over the unit ball, the unit sphere and the standard simplex. For $d=3$, extremal polynomial for $(x_1x_2x_3)^k$ on the ball and the sphere is found for $k=2$ and 4. For $d \ge 3$, a family of polynomials of the form $(x_1... x_d)^2 - p(x)$ is explicit given and proved to be the least deviation from zero for $d =3,4,5$, and it is conjectured to be the least deviation for all $d$.

Abstract:
Generalized translation operators for orthogonal expansions with respect to families of weight functions on the unit ball and on the standard simplex are studied. They are used to define convolution structures and modulus of smoothness for these regions, which are in turn used to characterize the best approximation by polynomials in the weighted $L^p$ spaces. In one variable, this becomes the generalized translation operator for the Gegenbauer polynomial expansions.

Abstract:
For weighted $L^1$ space on the unit sphere of $\RR^{d+1}$, in which the weight functions are invariant under finite reflection groups, a maximal function is introduced and used to prove the almost everywhere convergence of orthogonal expansions in $h$-harmonics. The result applies to various methods of summability, including the de la Vall\'ee Poussin means and the Ces\`aro means. Similar results are also established for weighted orthogonal expansions on the unit ball and on the simplex of $\RR^d$.

Abstract:
The second order partial difference equation of two variables $ \CD u:= A_{1,1}(x) \Delta_1 \nabla_1 u + A_{1,2}(x) \Delta_1 \nabla_2 u + A_{2,1}(x) \Delta_2 \nabla_1 u + A_{2,2}(x) \Delta_2 \nabla_2 u & \qquad \qquad \qquad \qquad + B_1(x) \Delta_1 u + B_2(x) \Delta_2 u = \lambda u, $ is studied to determine when it has orthogonal polynomials as solutions. We derive conditions on $\CD$ so that a weight function $W$ exists for which $W \CD u$ is self-adjoint and the difference equation has polynomial solutions which are orthogonal with respect to $W$. The solutions are essentially the classical discrete orthogonal polynomials of two variables.

Abstract:
Let $V$ be a set of isolated points in $\RR^d$. Define a linear functional $\CL$ on the space of real polynomials restricted on $V$, $\CL f = \sum_{x \in V} f(x)\rho(x)$, where $\rho$ is a nonzero function on $V$. Polynomial subspaces that contain discrete orthogonal polynomials with respect to the bilinear form $ = \CL(f g)$ are identified. One result shows that the discrete orthogonal polynomials still satisfy a three-term relation and Favard's theorem holds in this general setting.

Abstract:
A new approach is proposed for reconstruction of images from Radon projections. Based on Fourier expansions in orthogonal polynomials of two and three variables, instead of Fourier transforms, the approach provides a new algorithm for the computed tomography. The convergence of the algorithm is established under mild assumptions.