Schoenberg's polynomial B-splines. A short summary of applications
DOI:
https://doi.org/10.3103/S0735272710090013Keywords:
Schoenberg's B-splinesAbstract
A short summary of applications for Schoenberg's polynomial B-splines in statistical radio engineering and mathematical statistics is provided. Positive sides of Schoenberg's B-splines are specified; advisability of their wider application is noted. Literature where exact formulas for Schoenberg's B-splines of several minimal orders is listed.
References
E. Meijering, “A chronology of interpolation: from ancient astronomy to modern signal and image processing,” Proc. IEEE 90, No. 3, 319 (2002).
V. G. Alekseev, “New continuous low-pass filters,” Radiotekhnika, No. 4, 34 (1998).
V. G. Alekseev, “New analogue linear low-pass filter,” Radiotekhnika, No. 10, 143 (2005).
V. G. Alekseev and V. А. Sukhodoyev, “New analogue flat-topped and differentiating linear filters,” Elektomagnitnye Volny i Elektronnye Sistemy 11, No. 10, 21 (2006).
V. G. Alekseev, “Approximation of analogue filters with infinite impulse response,” Elektomagnitnye Volny i Elektronnye Sistemy 13, No. 1, 9 (2008).
Yu. А. Grebenko and Ch. Ya. Zei, “Complex active RC-filters on identical sections,” Radiotekhnika, No. 2, 61 (2008).
V. G. Alekseev and V. А. Sukhodoyev, “New sets of wavelet generating functions,” Izv. RAN. Fizika Atmosfery i Okeana 43, No. 5, 617 (2007).
Ye. I. Shapiro, “Non-parametric estimates of probability density in problems of processing the observations results,” Zarubezhnaya Radioelektronika, No. 2, 3 (1976).
V. G. Alekseev, “On non-parametric estimates of probability density and its derivatives,” Problemy Peredachi Informatsii 18, No. 2, 22 (1982).
K. B. Davis, “Mean integrated squared error properties of density estimates,” Annals of Statistics 5, No. 3, 530 (1977).
D. B. H. Cline, “Admissible kernel estimators of a multivariate density,” Annals of Statistics 16, No. 4, 1421 (1988).
V. G. Alekseev, “Non-parametric estimation of probability density and its derivatives. The L2 approach,” Avtometriya 43, No. 6, 39 (2007).
V. G. Alekseev, “Non-parametric spectral analysis of stationary random processes,” Avtometriya 36, No. 4, 131 (2000).
V. G. Alekseev and V. А. Sukhodoyev, “Welch-type estimation of spectral density. The case of continuous argument,” Avtometriya 45, No. 2, 22 (2009).
P. L. Butzer, M. Schmidt, E. L. Stark, and L. Vogt, “Central factorial numbers, their main properties and some applications,” Numerical Functional Analysis and Optimization 10, Nos. 5–6, 419 (1989).
M. Unser, A. Aldroubi, and M. Eden, “Polynomial spline signal approximations: filter design and asymptotic equivalence with Shannon’s sampling theorem,” IEEE Trans. Inf. Theory 38, No. 1, 95 (1992).
A. Aldroubi, M. Unser, and M. Eden, “Cardinal spline filters: stability and convergence to the ideal sinc interpolator,” Signal Processing 28, No. 2, 127 (1992).
J. J. Chen, A. K. Chan, and C. K. Chui, “A local interpolatory cardinal spline method for the determination of eigenstates in quantum–well structures with arbitrary potential profiles,” IEEE J. Quantum Electron. 30, No. 2, 269 (1994).
M. Unser and I. Daubechies, “On the approximation power of convolution–based least squares versus interpolation,” IEEE Trans. Signal Process. 45, No. 7, 1697 (1997).
Sh. А. Khashimov and K. Kh. Ubaidullayev, “On strong consistency of spline-estimation of probability density,” Uzbekskii Matematicheskii Zhurnal, No. 1, 60 (2000).
M. Unser, “Sampling—50 years after Shannon,” Proc. IEEE 88, No. 4, 569 (2000).
N. А. Strelkov, “Spline-trigonometric basis and their properties,” Matematicheskii Sbornik 192, No. 7, 125 (2001).
V. G. Alekseev, “Schoenberg’s B-splines and their application in radio engineering and neighboring disciplines,” Radiotekhnika, No. 12, 21 (2003).
H. Kano, M. Egerstedt, H. Nakata, and C. F. Martin, “B–splines and control theory,” Appl. Math. Optimiz. 145, No. 2–3, 263 (2003).
H. Kano, H. Nakata, and C. F. Martin, “Optimal curve fitting and smoothing using normalized uniform B–splines: a tool for studying complex systems,” Appl. Math. Comput. 169, No. 1, 96 (2005).
А. P. Kolesnikov, “Algebraic splines in locally concave up spaces,” Matematicheskie Zametki 77, No. 3, 339 (2005).
X. Liu, “Univariate and bivariate orthonormal splines and cardinal splines on compact supports,” J. Comput. Appl. Math. 195, No. 1–2, 93 (2006).
А. S. Zhuk and V. V. Zhuk, “On approximation of periodic functions with linear approximation methods,” Zapiski nauchnykh seminarov POMI 337, 134 (2006).
P. N. Podkur, “Building wavelets with scaling coefficient on the basis of B-splines,” Vestnik KemGU, No. 4, 19 (2006).
G. Apaydin, S. Seker, and N. Ari, “Weighted extended b–splines for one–dimentional electromagnetic problems,” Appl. Math. Comput. 190, No. 2, 1125 (2007).
H. Calgar and N. Calgar, “Fifth–degree B–spline solution for a fourth–order parabolic differential equation,” Appl. Math. Comput. 201, No. 1–2, 597 (2008).
V. G. Alekseev, “On estimation of mathematical expectation of periodically correlated random process,” Matematicheskie Zametki YaGU 15, No. 2, 3 (2008).
G. Vainicco, “Cardinal approximation of functions by splines on an interval,” Math. Modeling and Analysis 14, No. 1, 127 (2009).
R. Varga, Functional Analysis and Approximation Theory in Numerical Analysis (Mir, Moscow, 1974) [in Russian].
S. F. Svinyin, Basis Splines in Signals Samples Theory (Nauka, St. Petersburg, 2003) [in Russian].
C. K. Chui, An Introduction to Wavelets (Academic Press, San Diego, 1992; Mir, Moscow, 2001).
