Several Numerical Approximation Problems In Some Reproducing Kernel Spaces

In this paper, three numerical approximation problems are discussed in some reproducing kernel spaces.Firstly, in the reproducing kernel space, the spline function is discussed and its equivalent relations are gived; we prove that the 2-th differential operator interpolation spline function in the space can be expressed not only by the reproducing kernels but also by polygonal functions. So it is easy to prove its best approximation properties in theory and is convenient for numerical calculation in applications, too.Except for method of spline interpolation, MRA method are used to resolve numerical approximation problems, too. And by far, most theories of wavelet analysis are established in , but this method is rare to be known and exists defects in the reproducing kernel space with good properties.Then in the reproducing kernel space, isometric isomorphism of relation is founded between and by differential operator . And starting from the Haar basis , multi-resolution analysis inis gived by folding method; then multi-resolution analysis and orthonormal wavelet in are obtained through integral transform from to .Then the reproducing kernel space can be expressed by wavelet spaces. In the space , wavelet approximation formula and sampling formula are gived and these are simple and easy for calculation. It is a new attempt to establish MRA in the reproducing kernel space.This not only makes up existing researchful defects but also gives a better method to resolve numerical approximation problems and it further enriches the reproducing kernel theory and wavelet method . this provides more applied method for resolving numerical approximation problems in the space which is often used in the engineering. Finally, a tensor product space is constructed ,in which exists reproducing kernel. The explicit representation of the best interpolationoperator for the bivariate real functions is gived and the representation is not in polynomial form. Furthermore, the representation is convergent uniformly and the error of interpolation is degressive monotonically when rectangular grid knots are thickened infinitely. It provides some methods for finding two-dimensional interpolation formula to calculate double integral.