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Discrete smoothing splines and digital filtration. Theory and applications
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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).
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).
DOI : 10.21136/AM.1990.104385
Classification : 41A15, 65D07, 65D10, 65K10, 93E11, 93E14
Keywords: discrete smoothing spline CDS-spline; smoothing parameter; digital convolution filter; transfer function; sinusoidal wave; saw-like waves; rectangular pulse train 
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Hřebíček, Jiří; Šik, František; Veselý, Vítězslav. Discrete smoothing splines and digital filtration. Theory and applications. Applications of Mathematics, Tome 35 (1990) no. 1, pp. 28-50. doi: 10.21136/AM.1990.104385

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