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Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952

Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952
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The Besicovitch covering theorem and near-minimizers for the couple (L2, BV); pp. 29–33

(Full article in PDF format) doi: 10.3176/proc.2010.1.05


Authors

Irina Asekritova, Natan Kruglyak

Abstract

Let Ω be a rectangle in R2. A new algorithm for the construction of a near-minimizer for the couple (L2(Ω), BV(Ω)) is presented. The algorithm is based on the Besicovitch covering theorem and analysis of local approximations of the given function f L2(Ω).

Keywords

covering theorems, near-minimizers.

References

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  2. Brudnyi , Yu. A. and Krugljak , N. Interpolation Functors and Interpolation Spaces 1. North Holland , Amsterdam , 1991.

  3. Kirsch , A. An Introduction to the Mathematical Theory of Inverse Problems. Springer , New York , 1996.

  4. Chan , T. and Shen , J. Image Processing and Analysis: Variational , PDE , Wavelet , and Stochastic Methods. SIAM , Philadelphia , 2005.

  5. Meyer , Y. Oscillating Pattern in Image Processing and Nonlinear Evolution Equations. University Lecture Series , AMS Providence , 2002 , 22.

  6. Bechler , P. , DeVore , R. , Kamot , A. , Petrova , G. , and Wojtaszczyk , P. Greedy wavelet projections are bounded on BV. Trans. AMS , 2007 , 359 , 619–635.
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  7. Cohen , A. , DeVore , R. , Petrushev , P. , and Xu , H. Nonlinear approximation and the space BV(R 2). Amer. J. Math. , 1999 , 121 , 587–628.
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  8. Cohen , A. , Dahmen , W. , Daubechies , I. , and DeVore , R. Harmonic analysis of the space BV. Rev. Mat. Iberoamericana , 2003 , 19 , 235–263.

  9. Daubechies , I. , Teschke , G. , and Vese , L. Iteratively solving linear inverse problems with general convex constraints. Inverse Probl. Imaging , 2007 , 1 , 29–46.

10. Kruglyak , N. Smooth analogs of Calderon–Zygmund decompositions , quantitative covering theorems and K-functional for the couple (LpWqk). Algebra i Analiz , 1996 , 8 , 110–160 (in Russian); English translation in St.-Petersburg Math. Journal , 1997 , 8 , 617–649.

11. Kruglyak , N. The K-functional and Calderón–Zygmund type decompositions. Contemporary Math. , 2007 , 446 , 183–194.

12. Brudnyi , Yu. A. Spaces determined by local approximations. Trudy Mosk. Matem. Ob-va , 1971 , 24 , 69–132.

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doi:10.1017/S0305004100022660

15. de Guzman , M. Differentiation of Integrals in Rn. Lecture Notes in Mathematics , Springer-Verlag , New York , 1975 , 481.
 
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Current Issue: Vol. 68, Issue 4, 2019




Publishing schedule:
No. 1: 20 March
No. 2: 20 June
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No. 4: 20 December