Schauder decomposition and its application to integral equations of the second kind (Q796792)

The rates of convergence of Schauder decompositions in \(L_ p\) and \(W_ p^{(n)}\) (the Sobolev space) are established. The existence of such decompositions have been established previously in Num. Funct. Anal. Optimization 4, 45-59 (1981; Zbl 0483.46013) (\textit{H. Kaneko} and \textit{W. Ruckle}). Under suitable assumptions, if \(\{P_ k\}\) and \(\{Q^ n_ k\}\) are Schauder decompositions for \(L_ p\) and \(W_ p^{(n)}\) respectively then it is shown that \[ \| f-P_ kf\|_{\infty}=O(\omega(f,1/3^ k))\quad and\quad \| f-Q^ n_ kf\|_{\infty}=O(\omega(f^{(n)},1/3^ k)). \] An application is made to integral equations of the second kind.