The Rate of Convergence for the Finite Element Method

Summary:Let $\{\omega_y\}$ be a system of infinitely smooth rapidly decreasing functions and $\eta (h)$ a certain increasing function, $\eta (0)=0$. Then the approximation sought in the form $\sum c_k\omega_{\eta(h)}((x/h-k)\eta(h))$ is universal, i.e., for any approximated function $f$, the system $\{\omega_y\}$ of hill functions gives the best possible order of approximation limited only by the smoothness of $f$. Moreover, the system $\{\omega_y\}$ can be chosen so that the Fourier transform of $\omega_y$ has zeros at the points $\pm2\pi j/y; j=1,\ldots, J$. As a consequence, the error of the approximation decreases