Bounded Area On The Radial Basis Function Research Of Bernstein Inequality

Radial basis function (RBF) interpolation is one of the most powerful tools toapproximate multivariate functions. Theoretical analysis, for example, the Sobolev-typeerror estimates, this method has been extensively studied for several decades. It isalso used to solve partial differential equations numerically, called, RBF-based meshlessmethod.The same as the inverse inequality of polynomial approximation, Bernstein-Typeinequality also plays an important role in RBF interpolation theory and meshlessmethod theoretical analysis. In practical applications, functions are generally defined ona bounded domain or we just care about the values on a bounded domain. In this paper,we present Bernstein-Type inequality on open bounded domain by using of theband-limited functions and scale kernel.For RBF interpolation, because of uncertainty principle, the smaller the density ofdata points, the larger condition number of interpolation matrix，even singular. It willlead to the equations serious distortion, therefore, the stability of the RBF interpolationmatrix is an important work in this paper. We will study the condition number for theunsymmetric collocation on the boundary value problem to obtain the interpolationmatrix the relationship between the condition numbers and the data point density.According to this conclusion and the property of the Native space, we can obtain theinverse inequality inR n. The numerical examples show this inverse inequality isfeasible, and the results are satisfying.