Geodesic
Universal approximation by systems of hill functions
Applications of Mathematics, Tome 19 (1974) no. 6, pp. 403-436
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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.
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.
DOI : 10.21136/AM.1974.103558
Classification : 41A30, 65N35 
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Segeth, Karel. Universal approximation by systems of hill functions. Applications of Mathematics, Tome 19 (1974) no. 6, pp. 403-436. doi: 10.21136/AM.1974.103558

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