Best Polynomial Approximation on the Unit Sphere and the Unit Ball

نویسنده

  • YUAN XU
چکیده

This is a survey on best polynomial approximation on the unit sphere and the unit ball. The central problem is to describe the approximation behavior of a function by polynomials via smoothness of the function. A major effort is to identify a correct gadget that characterizes smoothness of functions, either a modulus of smoothness or a Kfunctional, the two of which are often equivalent. We will concentrate on characterization of best approximations, given in terms of direct and converse theorems, and report several moduli of smoothness and K-functionals, including recent results that give a fairly satisfactory characterization of best approximation by polynomials for functions in L spaces, the space of continuous functions, and Sobolev spaces.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Moduli of smoothness and approximation on the unit sphere and the unit ball

A new modulus of smoothness based on the Euler angles is introduced on the unit sphere and is shown to satisfy all the usual characteristic properties of moduli of smoothness, including direct and inverse theorem for the best approximation by polynomials and its equivalence to a K-functional, defined via partial derivatives in Euler angles. It also leads to new moduli of smoothness and K-functi...

متن کامل

Best Approximation of Monomials in Several Variables

For a monomial x in d variables, the problem of best approximation to x by polynomials of lower degrees is studied on the unit sphere, the unit ball and the standard simplex. For the uniform norm we discuss what is known in d = 2, for which complete solutions are known, and in d ≥ 3 for which only a few cases have been successfully solved. For the L norm, we present the complete solution, inclu...

متن کامل

Analysis on the Unit Ball and on the Simplex

Many results on the unit ball and those on the simplex can be deduced from each other or from the corresponding results on the unit sphere. The areas in which such a connection appears include orthogonal polynomials, approximation, cubature formulas and polynomial interpolation. We explain this phenomenon in some detail.

متن کامل

On Polynomial Interpolation on the Unit Ball

Polynomial interpolation on the unit ball of R d has a unique solution if the points are located on several spheres inside the ball and the points on each sphere solves the corresponding interpolation problem on the sphere. Furthermore, the solution can be computed in a recursive way.

متن کامل

Weighted Approximation of Functions on the Unit Sphere

The direct and inverse theorems are established for the best approximation in the weighted L space on the unit sphere of R, 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 ...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2014