# Discrete smoothing splines and digital filtration. Theory and applications

Jiří Hřebíček; František Šik; Vítězslav Veselý

Aplikace matematiky (1990)

- Volume: 35, Issue: 1, page 28-50
- ISSN: 0862-7940

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topHřebíček, Jiří, Šik, František, and Veselý, Vítězslav. "Discrete smoothing splines and digital filtration. Theory and applications." Aplikace matematiky 35.1 (1990): 28-50. <http://eudml.org/doc/15608>.

@article{Hřebíček1990,

abstract = {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 $\mathcal \{A\}$, $\mathcal \{L\}$ 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 =\lbrace t_i=ih\rbrace ^\{n-1\}_\{i=0\}$$T=nh$. The smoothed data vector $y$ is then $y=\lbrace \sigma _\alpha (t_i)\rbrace ^\{n-1\}_\{i=0\}$. The other operation gives $y\in E^n$ computed by $\mathbf \{y=h*r\}$, where $\mathbf \{*\}$ 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).},

author = {Hřebíček, Jiří, Šik, František, Veselý, Vítězslav},

journal = {Aplikace matematiky},

keywords = {discrete smoothing spline CDS-spline; smoothing parameter; digital convolution filter; transfer function; sinusoidal wave; saw-like waves; rectangular pulse train; smoothing parameter; digital convolution filter; discrete smoothing splines; digital filtration; transfer function; sinusoidal wave; saw-like waves; rectangular pulse train},

language = {eng},

number = {1},

pages = {28-50},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {Discrete smoothing splines and digital filtration. Theory and applications},

url = {http://eudml.org/doc/15608},

volume = {35},

year = {1990},

}

TY - JOUR

AU - Hřebíček, Jiří

AU - Šik, František

AU - Veselý, Vítězslav

TI - Discrete smoothing splines and digital filtration. Theory and applications

JO - Aplikace matematiky

PY - 1990

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 35

IS - 1

SP - 28

EP - 50

AB - 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 $\mathcal {A}$, $\mathcal {L}$ 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 =\lbrace t_i=ih\rbrace ^{n-1}_{i=0}$$T=nh$. The smoothed data vector $y$ is then $y=\lbrace \sigma _\alpha (t_i)\rbrace ^{n-1}_{i=0}$. The other operation gives $y\in E^n$ computed by $\mathbf {y=h*r}$, where $\mathbf {*}$ 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).

LA - eng

KW - discrete smoothing spline CDS-spline; smoothing parameter; digital convolution filter; transfer function; sinusoidal wave; saw-like waves; rectangular pulse train; smoothing parameter; digital convolution filter; discrete smoothing splines; digital filtration; transfer function; sinusoidal wave; saw-like waves; rectangular pulse train

UR - http://eudml.org/doc/15608

ER -

## References

top- P. M. Anselone P.-J. Laurent, 10.1007/BF02170998, Num. Math. 12 (1968) No. 1, 66-82. (1968) MR0249904DOI10.1007/BF02170998
- P. Bečička J. Hřebíček F. Šik, Numerical analysis of smoothing splines, (Czech). Proceed. 9-th Symposium on Algorithms ALGORITMY 87, JSMF, Bratislava. 1987, 22-24. (1987)
- K. Böhmer, Spline-Funktionen, Teubner, Stuttgart, 1974. (1974) MR0613676
- E. O. Brigham, The Fast Fourier Transform, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1974. (1974) Zbl0375.65052
- C. S. Burrus T. W. Parks, DFT/FFT and Convolution Algorithms, Wiley Interscience, 1985. (1985)
- P. Craven G. Wahba, 10.1007/BF01404567, Numer. Math. 31 (1979), 377-403. (1979) MR0516581DOI10.1007/BF01404567
- D. F. Elliot K. R. Rao, Fast transforms. Algorithms, Analyses, Applications, Acad. Press, New York, London, 1982. (1982) MR0696936
- W. Gautschi, 10.1007/BF01406676, Num. Math. 18 (1972), 373-400. (1972) Zbl0231.65101MR0305641DOI10.1007/BF01406676
- M. H. Gutknecht, 10.1007/BF01400173, Num. Math. 51 (1987), 615-629. (1987) Zbl0639.65079MR0914342DOI10.1007/BF01400173
- J. Hřebíček F. Šik V. Veselý, Digital convolution filters and smoothing splines, Proceed. 2nd ISNA (I. Marek, ed.), Prague 1987, Teubner, Leipzig, 1988, 187-193. (1987) MR1171704
- J. Hřebíček F. Šik V. Veselý, How to choose the smoothing parameter of a periodic smoothing spline, (to appear).
- J. Hřebíček F. Šik P. Švenda V. Veselý, Smoothing splines and digital filtration, Research Report, Czechoslovak Academy of Sciences, Institute of Physical Metallurgy, Brno, 1987. (1987)
- L. V. Kantorovič V. I. Krylov, Approximate methods of higher analysis, (in Russian). 4. ed. Moskva, 1952. (1952) MR0106537
- P. J. Laurent, Approximation et Optimisation, Hermann, Paris, 1972. (1972) Zbl0238.90058MR0467080
- F. Locher, 10.1090/S0025-5718-1981-0628704-2, Math. Comput. 37 (1981) No. 156, 403 - 416. (1981) Zbl0517.42004MR0628704DOI10.1090/S0025-5718-1981-0628704-2
- M. Marcus H. Minc, A survey of matrix theory and matrix inequalities, Boston 1964 (Russian translation, Nauka, Moskva, 1972). (1964) MR0349699
- H. J. Nussbaumer, Fast Fourier Transform and Convolution Algorithms, 2nd ed., Springer, Berlin, Heidelberg, New York, 1982. (1982) MR0606376
- V. A. Vasilenko, Spline-Functions: Theory, Algorithms, Programs, (in Russian). Nauka, Novosibirsk, 1983. (1983) Zbl0529.41013MR0721970
- 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. (1988)

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