| Title:
|
Discrete smoothing splines and digital filtration. Theory and applications (English) |
| Author:
|
Hřebíček, Jiří |
| Author:
|
Šik, František |
| Author:
|
Veselý, Vítězslav |
| Language:
|
English |
| Journal:
|
Aplikace matematiky |
| ISSN:
|
0373-6725 |
| Volume:
|
35 |
| Issue:
|
1 |
| Year:
|
1990 |
| Pages:
|
28-50 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Two universally applicable smoothing operations adjustable to meet the specific properties of the given smoothing problem are widely used: 1. Smoothing splines and 2. Smoothing digital convolution filters. The first operation is related to the data vector $r={(r_0,..., r_{n-1})}^T$ with respect to the operations $\Cal{A}$, $\Cal{L}$ and to the smoothing parameter $\alpha$. The resulting function is denoted by $\sigma_\alpha(t)$. The measured sample $r$ is defined on an equally spaced mesh $\Delta=\{t_i=ih\}^{n-1}_{i=0}$ $T=nh$. The smoothed data vector $y$ is then $y=\{\sigma_\alpha(t_i)\}^{n-1}_{i=0}$. The other operation gives $y\in E^n$ computed by $\bold {y=h*r}$, where $\bold *$ stands for the discrete convolution, the running weighted mean by $h$. The main aims of the present contribution: to prove the existence of close interconnection between the two smoothing approaches (Cor. 2.6 and [11]), to develop the transfer function, which characterizes the smoothing spline as a filter in terms of $\alpha$ and $\lambda_{ik}$ (the eigenvalues of the discrete analogue of $Cal {L}$) (Th. 2.5), to develop the reduction ratio between the original and the smoothed data in the same terms (Th. 3.1). (English) |
| Keyword:
|
discrete smoothing spline CDS-spline |
| Keyword:
|
smoothing parameter |
| Keyword:
|
digital convolution filter |
| Keyword:
|
transfer function |
| Keyword:
|
sinusoidal wave |
| Keyword:
|
saw-like waves |
| Keyword:
|
rectangular pulse train |
| MSC:
|
41A15 |
| MSC:
|
65D07 |
| MSC:
|
65D10 |
| MSC:
|
65K10 |
| MSC:
|
93E11 |
| MSC:
|
93E14 |
| idZBL:
|
Zbl 0704.65005 |
| idMR:
|
MR1039409 |
| DOI:
|
10.21136/AM.1990.104385 |
| . |
| Date available:
|
2008-05-20T18:38:18Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/104385 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[11] J. Hřebíček F. Šik V. Veselý: How to choose the smoothing parameter of a periodic smoothing spline.(to appear). |
| Reference:
|
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| Reference:
|
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|
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| Reference:
|
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|
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|
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| Reference:
|
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| Reference:
|
[19] J. Hřebíček F. Šik V. Veselý: Smoothing by discrete splines and digital convolution filters.(Czech). Proceed Conf. Numer. Methods in the Physical Metallurgy (J. Hřebíček, ed.) Blansko 1988, ÚFM ČSAV Brno 1988, 62-70. |
| . |
