| Title:
|
Universal approximation by systems of hill functions (English) |
| Author:
|
Segeth, Karel |
| Language:
|
English |
| Journal:
|
Aplikace matematiky |
| ISSN:
|
0373-6725 |
| Volume:
|
19 |
| Issue:
|
6 |
| Year:
|
1974 |
| Pages:
|
403-436 |
| Summary lang:
|
English |
| Summary lang:
|
Czech |
| . |
| Category:
|
math |
| . |
| 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. () |
| MSC:
|
41A30 |
| MSC:
|
65N35 |
| idZBL:
|
Zbl 0305.41011 |
| idMR:
|
MR0388812 |
| DOI:
|
10.21136/AM.1974.103558 |
| . |
| Date available:
|
2008-05-20T18:00:09Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/103558 |
| . |
| Reference:
|
[1] I. Babuška: Approximation by hill functions.Comment. Math. Univ. Carolinae 11 (1970), 787-811. MR 0292309 |
| Reference:
|
[2] I. Babuška: The rate of convergence for the finite element method.SIAM J. Numer. Anal. 8 (1971), 304-315. MR 0287715, 10.1137/0708031 |
| Reference:
|
[3] I. Babuška J. Segethová K. Segeth: Numerical experiments with the finite element method I.Tech. Note BN-669, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, August 1970. |
| Reference:
|
[4] G. Fix G. Strang: Fourier analysis of the finite element method in Ritz-Galerkin theory.Studies in Appl. Math. 48 (1969), 265-273. MR 0258297, 10.1002/sapm1969483265 |
| Reference:
|
[5] J. L. Lions E. Magenes: Problèmes aux limites non homogènes et applications.Vol. 1. Dunod, Paris 1968. MR 0247243 |
| Reference:
|
[6] J. Nečas: Les méthodes directes en théorie des équations elliptiques.Academia, Prague 1967. MR 0227584 |
| Reference:
|
[7] K. Segeth: Universal approximation by hill functions.Czechoslovak Math. J. 22 (1972), 612-640. Zbl 0247.41011, MR 0310502 |
| Reference:
|
[8] K. Segeth: A remark on a class of universal hill functions.Acta Univ. Carolinae-Math. et Phys. 15 (1974), No. 1 - 2, to appear. MR 0390598 |
| Reference:
|
[9] G. Strang G. J. Fix: An analysis of the finite element method.Prentice-Hall, Englewood Cliffs, N. J. 1973. MR 0443377 |
| Reference:
|
[10] K. Yosida: Functional analysis.Academic Press, New York-London 1965. Zbl 0126.11504, MR 0180824 |
| . |
