Abstract:
We introduce a new norm and a new K-functional (;),. Using this K-functional, direct and inverse approximation theorems for the Baskakov operators with the Jacobi-type weight are obtained in this paper.

Abstract:
We consider the best approximation by Jackson-Matsuoka polynomials in the weighted space on the unit sphere of ？. Using the relation between -functionals and modulus of smoothness on the sphere, we obtain the direct and inverse estimate of approximation by these polynomials for the ？-spherical harmonics.

Abstract:
We consider the classes of periodic functions with formal self-adjoint linear differential operators (？), which include the classical Sobolev class as its special case. Using the iterative method of Buslaev, with the help of the spectrum of linear differential equations, we determine the exact values of Bernstein width of the classes (？) in the space for 1<≤<∞.

Abstract:
We consider the classes of periodic functions with formal self-adjoint linear differential operators , which include the classical Sobolev class as its special case. With the help of the spectral of linear differential equations, we find the exact values of Bernstein -width of the classes in the for .

Abstract:
We consider some classes of 2 €-periodic convolution functions B p, and K p, which include the classical Sobolev class as a special case. With the help of the spectra of nonlinear integral equations, we determine the exact values of Bernstein n-width of the classes B p, K p in the space Lp for 1 Keywords

Abstract:
We consider the classes of periodic functions with formal self-adjoint linear differential operators Wp( ￠ ’r), which include the classical Sobolev class as its special case. With the help of the spectral of linear differential equations, we find the exact values of Bernstein n-width of the classes Wp( ￠ ’r) in the Lp for 1 Keywords

Abstract:
We consider some classes of -periodic convolution functions , and , which include the classical Sobolev class as a special case. With the help of the spectra of nonlinear integral equations, we determine the exact values of Bernstein -width of the classes , in the space for .

Abstract:
This paper studies a class of so-called linear semi-infinite polynomial programming (LSIPP) problems. It is a subclass of linear semi-infinite programming problems whose constraint functions are polynomials in parameters and index sets are basic semialgebraic sets. When the index set of an LSIPP problem is compact, a convergent hierarchy of semidefinite programming (SDP) relaxations is constructed under the assumption that the Slater condition and the Archimedean property hold. When the index set is noncompact, we use the technique of homogenization to equivalently convert the LSIPP problem into compact case under some generic assumption. Consequently, a corresponding hierarchy of SDP relaxations for noncompact LSIPP problems is obtained. We apply this relaxation approach to the special LSIPP problem reformulated from a polynomial optimization problem. A new SDP relaxation method is derived for solving the class of polynomial optimization problems whose objective polynomials are stably bounded from below on noncompact feasible sets.

Abstract:
Semidefinite programs are an important class of convex optimization problems. It can be solved efficiently by SDP solvers in Matlab, such as SeDuMi, SDPT3, DSDP. However, since we are running fixed precision SDP solvers in Matlab, for some applications, due to the numerical error, we can not get good results. SDPTools is a Maple package to solve SDP in high precision. We apply SDPTools to the certification of the global optimum of rational functions. For the Rumps Model Problem, we obtain the best numerical results so far.