# Article

Full entry | PDF   (3.2 MB)
Keywords:
discrete smoothing spline CDS-spline; smoothing parameter; digital convolution filter; transfer function; sinusoidal wave; saw-like waves; rectangular pulse train
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).
References:
[1] P. M. Anselone P.-J. Laurent: A general method for the construction of interpolating or smoothing spline functions. Num. Math. 12 (1968) No. 1, 66-82. DOI 10.1007/BF02170998 | MR 0249904
[2] 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.
[3] K. Böhmer: Spline-Funktionen. Teubner, Stuttgart, 1974. MR 0613676
[4] E. O. Brigham: The Fast Fourier Transform. Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1974. Zbl 0375.65052
[5] C. S. Burrus T. W. Parks: DFT/FFT and Convolution Algorithms. Wiley Interscience, 1985.
[6] P. Craven G. Wahba: Smoothing Noisy Data with Spline Functions. Numer. Math. 31 (1979), 377-403. DOI 10.1007/BF01404567 | MR 0516581
[7] D. F. Elliot K. R. Rao: Fast transforms. Algorithms, Analyses, Applications. Acad. Press, New York, London, 1982. MR 0696936
[8] W. Gautschi: Attenuation Factors in Practical Fourier Analysis. Num. Math. 18 (1972), 373-400. DOI 10.1007/BF01406676 | MR 0305641 | Zbl 0231.65101
[9] M. H. Gutknecht: Attenuation factors in multivariate Fourier analysis. Num. Math. 51 (1987), 615-629. DOI 10.1007/BF01400173 | MR 0914342 | Zbl 0639.65079
[10] 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. MR 1171704
[11] J. Hřebíček F. Šik V. Veselý: How to choose the smoothing parameter of a periodic smoothing spline. (to appear).
[12] 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.
[13] L. V. Kantorovič V. I. Krylov: Approximate methods of higher analysis. (in Russian). 4. ed. Moskva, 1952. MR 0106537
[14] P. J. Laurent: Approximation et Optimisation. Hermann, Paris, 1972. MR 0467080 | Zbl 0238.90058
[15] F. Locher: Interpolation on uniform meshes by the translates of one function and related attenuation factors. Math. Comput. 37 (1981) No. 156, 403 - 416. DOI 10.1090/S0025-5718-1981-0628704-2 | MR 0628704 | Zbl 0517.42004
[16] M. Marcus H. Minc: A survey of matrix theory and matrix inequalities. Boston 1964 (Russian translation, Nauka, Moskva, 1972). MR 0349699
[17] H. J. Nussbaumer: Fast Fourier Transform and Convolution Algorithms. 2nd ed., Springer, Berlin, Heidelberg, New York, 1982. MR 0606376
[18] V. A. Vasilenko: Spline-Functions: Theory, Algorithms, Programs. (in Russian). Nauka, Novosibirsk, 1983. MR 0721970 | Zbl 0529.41013
[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.

Partner of