The Real Axis Sobolev Kind Of Approximation Of The Probolem

With the development of modern science,the approximation theory of is an important branch of modern mathematics, which contains increasingly extensive content and combines with other subjects deeply. It not only closely related to functional analysis, differential equation, algebra, numerical analysis, harmonic analysis but also wavelet analysis research.Meanwhile, it has become the foundation of computational mathematics applied mathematics and optimization theory.Function approximation have a profoundly theoretical background and wide application prospect.The theoretical demand of numerical analysis,harmonic analysis and wavelet theory has been promotions the research and the development of approximation theory of function. In this paper,the approximation tool is the polynomial spline,and the approximation problem in Sobolev classes Wρ'(R)defined on the real line was studied. The approximation error was studied when r=1,p=1 p=∞and r=N, p=2. If n∈N,f∈W11(R),then ||f-Vn(f)||1≤1/2n。If n∈N,f∈W∞1(R),then ||f-Vn(f)||∞≤1/2n。If n∈N,f∈W2r(R),then ||f-S2r-1,w||≤π-rw-r.