Abstract
We expand upon the known results on sharp linear Fourier methods of approximation where the approximation is the best in terms of both rate and constant among all polynomial procedures of approximation. So far these results have been studied due to their mathematical beauty rather than their practical importance. In this paper we show that they are the core mathematics underlying best statistical methods of solving noisy ill-posed problems. In particular, we suggest a procedure for recovery of noisy blurred signals based on samples of small sizes where a traditional statistics concludes that the complexity of such a setting makes the problem not worthy of a further study. Thus, we present a problem where a combination of the classical approximation theory and statistics leads to interesting practical results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
N. I. Akhiezer and M. G. Krein, On the best approximation of periodic functions, Dokl. Akad. Nauk SSSR 15, 107-111 (1937).
V. F. Babenko and S. A. Pichugov, Best linear approximation of certain classes of differentiable periodic functions, Math. Notes 27, 325-329 (1980).
S. N. Bernstein, On the best approximation of |x|p by means of polynomials of extremely high degree, Izv. Akad. Nauk SSSR 2, 169-180 (1938).
R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer-Verlag, New York, 1993.
V. K. Dzyadyk, On best approximation of the class of periodic functions with bounded sth derivative (0 < s < 1), Izv. Akad. Nauk SSSR 17, 135-162 (1953).
V. K. Dzyadyk, Best approximation on classes of periodic functions defined by kernels which are integrals of absolutely monotonic functions, Izv. Akad. Nauk SSSR 23, 933-950 (1959).
S. Efromovich and M. S. Pinsker, Learning algorithm for nonparametric filtering, Autom. Remote Control 11, 1434-1440 (1984).
S. Efromovich, Density estimation for the case of supersmooth measurement errors, J. Am. Stat. Assoc. 92, 526-535 (1997).
S. Efromovich, On nonparametric regression for iid observations in a general setting, Ann. Stat. 24, 1126-1143 (1996).
S. Efromovich and M. Pinsker, Sharp-optimal and adaptive estimator for heteroscedastic nonparametric regression, Stat. Sin. 6, 925-942 (1996).
M. Ermakov, Minimax estimation of the solution of an ill-posed convolution type problem, Probl. Inf. Transm. 25, 191-200 (1989).
J. Fan and Y. K. Truong, Nonparametric regression with errors in variables, Ann. Stat. 21, 1900-1925 (1993).
J. Favard, Sur l'approximation des functions periodiques par des polynomial-trigonometriques, Bull. Sci. Math. 61, 209-224, 243–256 (1936).
M. I. Ganzburg, On the best harmonic approximation of convolutions in L 1 and L ∞, Sov. Math. (Izv. VUZ) 29, 22-27 (1985).
Y. V. Ingster, Minimax testing nonparametric hypotheses about density distributions in L p metrics, Theory Probab. Its Appl. 31, 384-389 (1986).
A. N. Kolmogorov, Zur Grossenordung des restgliedes Fourierscher Reihen differenzierbarer Funktionen, Ann. Math. 36, 107-110 (1935).
N. Korneichuk, Exact Constants in Approximation Theory, Cambridge Univ. Press, New York, 1991.
A. Korostelev and A. Tsybakov, Minimax Theory of Image Reconstruction, Springer-Verlag, New York, 1993.
E. L. Lehmann, Theory of Point Estimation, Wadsworth and Brooks, Pacific Grove, 1983.
S. M. Nikolskii, Mean approximation of functions by trigonometric polynomials, Izv. Akad. Nauk SSSR 10, 207-256 (1946).
R. J. Serfling, Approximations Theorems of Mathematical Statistics, Wiley, New York, 1980.
Yun-Shen Sun, On the best approximation of periodic differentiable functions by trigonometric polynomials, Izv. Akad. Nauk SSSR 25, 143-152 (1961).
V. M. Tikhomirov, Some Questions of Approximation Theory, Nauka, Moscow, 1976.
A. F. Timan, Theory of Approximation of Functions of a Real Variable, Pergamon Press, New York, 1963.
R. S. Varga and A. V. Carpenter, On the Bernstein conjecture in approximation theory, Constr. Approx. 1, 333-348 (1985).
A. Zygmund, Trigonometric Series, Cambridge Press, Cambridge, 1959.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Efromovich, S., Ganzburg, M. Best Fourier Approximation and Application in Efficient Blurred Signal Reconstruction. Journal of Computational Analysis and Applications 1, 43–62 (1999). https://doi.org/10.1023/A:1022666503405
Issue Date:
DOI: https://doi.org/10.1023/A:1022666503405