(The) study of the best polynomial approximation in the sobolev space

Abstract

In this thesis, we deal with the behavior of the best polynomial approximation in the inner product spaces $W^{N,2}[-1,1]$ and $W^{1,2}[0,∞ ;e^{-x}].$ We consider two different inner products in $W^{N,2}[-1,1]$, $W^{1,2}[0,∞ ;e^{-x}]$. We denote the best polynomial approximation in two different inner products by $B_{n,α}(x),$ $B_n(x),$ respectively in both inner product spaces. Then we have $\lim_{α→∞}B_{n,α(x)=B_n(x)$ in both inner product spaces. We see that the error of the best polynomial approximation alternates the sign. And the difference of the two best polynomial approximations in different inner products forms a Sobolev orthogonal polynomial system, besides, in $W^{N,2}[-1,1]$, which is complete with respect to the norm which comes from the inner product.