Discrete smoothing splines and digital filtration. Theory and applications
Hřebíček, Jiří ; Šik, František ; Veselý, Vítězslav
Applications of Mathematics, Tome 35 (1990), p. 28-50 / Harvested from Czech Digital Mathematics Library
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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).

Publié le : 1990-01-01
Classification:  41A15,  65D07,  65D10,  65K10,  93E11,  93E14 
@article{104385,
     author = {Ji\v r\'\i\ H\v reb\'\i \v cek and Franti\v sek \v Sik and V\'\i t\v ezslav Vesel\'y},
     title = {Discrete smoothing splines and digital filtration. Theory and applications},
     journal = {Applications of Mathematics},
     volume = {35},
     year = {1990},
     pages = {28-50},
     zbl = {0704.65005},
     mrnumber = {1039409},
     language = {en},
     url = {http://dml.mathdoc.fr/item/104385}
}

Hřebíček, Jiří; Šik, František; Veselý, Vítězslav. Discrete smoothing splines and digital filtration. Theory and applications. Applications of Mathematics, Tome 35 (1990) pp. 28-50. http://gdmltest.u-ga.fr/item/104385/

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