Complutense University Library

Exact minimizer for the couple (L(infinity), BV) and the one-dimensional analogue of the Rudin-Osher-Fatemi model

Cobos, Fernando and Kruglyak, Natan (2011) Exact minimizer for the couple (L(infinity), BV) and the one-dimensional analogue of the Rudin-Osher-Fatemi model. Journal of Approximation Theory, 163 (4). pp. 481-490. ISSN 0021-9045

[img] PDF
Restricted to Repository staff only until 2020.

195kB

Official URL: http://www.sciencedirect.com/science/article/pii/S0021904510002030

View download statistics for this eprint

==>>> Export to other formats

Abstract

We provide a simple algorithm that constructs an exact minimizer for the E-functional
E(t, f ; L∞, BV) = inf
‖g‖L∞≤t
‖ f − g‖BV .
Here L∞, BV stand for the space of bounded functions and the space of functions with bounded variation
on the interval [a, b], respectively. As a corollary we obtain the following formula for the K-functional
K(N, f ; BV, L∞) v sup
a≤x0≤···≤xN≤b
−N
i=1
| f (xi ) − f (xi+1)|.
We also discussed the connection between the results and the Rudin–Osher–Fatemi denoising model.

Item Type:Article
Uncontrolled Keywords:Exact minimizer; E-functional; K-functional; Rudin-Osher-Fatemi model; L-2; Mathematics
Subjects:Sciences > Mathematics > Mathematical analysis
ID Code:14900
References:

[1] I. Asekritova, N. Kruglyak, The Besicovitch covering theorem and near-minimizers for the couple (L2, BV), Proc.

Estonian Acad. Sci. 59 (2010) 29–33.

[2] P. Bechler, R. DeVore, A. Kamot, G. Petrova, P. Wojtaszczyk, Greedy wavelet projections are bounded on BV,

Trans. Amer. Math. Soc. 359 (2007) 619–635.

[3] J. Bergh, J. L¨ofstr¨om, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin, 1976.

[4] J. Bergh, J. Peetre, On the spaces Vp(0 < p < ∞), Boll. Unione Mat. Ital. 10 (1974) 632–648.

[5] Yu.A. Brudnyi, N. Krugljak, Interpolation Functors and Interpolation Spaces, vol. 1, North Holland, Amsterdam,

1991.

[6] T. Chan, J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods, SIAM,

Philadelphia, 2005.

[7] A. Cohen, R. DeVore, P. Petrushev, H. Xu, Nonlinear approximation and the space BV(R2), Amer. J. Math. 121

(1999) 587–628.

[8] A. Hess, G. Pisier, The Kt -functional for the interpolation couple (L∞(dμ; L1(dν)), L∞(dν; L1(dμ))), Q. J.

Math. (Oxford) 46 (1995) 333–344.

[9] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer, 1996.

[10] P. Nilsson, J. Peetre, On the K-functional between L1 and L2 and some other K-functionals, J. Approx. Theory 48

(1986) 322–327.

[11] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.

Deposited On:19 Apr 2012 09:16
Last Modified:22 Oct 2013 14:48

Repository Staff Only: item control page