Multivariate n-term rational and piecewise polynomial approximation

Journal of Approximation Theory

Volumn 121, Issue 1, 2003, Pages 158-197

  Petrushev, Pencho ^a  

^a UNIVERSITY OF SOUTH CAROLINA   (United States)

Author keywords

Multivariate piecewise polynomial approximation; Nonlinear approximation; Rational approximation

Indexed keywords

EID: 0037357249     PISSN: 00219045     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0021-9045(02)00060-6     Document Type: Review

References (24)

1. J. Bergh, J. Löfström, Interpolation spaces: An introduction, in: Grundlehren der Mathematischen Wissenschaften, Vol. 223, Springer, Berlin, New York, 1976.
2. Yu. Brudnyi, Approximation of functions of n-variables by quasi-polynomials, Math. USSR Izv. 4 (1970) 568-586.
3. R. Coifman, M. Wickerhauser, Entropy based algorithms for best basis selection, IEEE Trans. Inform. Theory 32 (1992) 712-718.
4. 2), Amer. J. Math. 121 (1999) 587-628.
5. R. DeVore, B. Jawerth, V. Popov, Compression of wavelet decompositions, Amer. J. Math. 114 (1992) 737-785.
6. R. DeVore, G. Lorentz, Constructive Approximation, Vol. 303, Springer Grundlehren, Heidelberg, 1993.
7. d), in: A.A. Gonchar, E.B. Saff (Eds.), Progress in Approximation Theory, Springer, New York, 1992, pp. 261-283.
8. R. DeVore, V.A. Popov, Interpolation of Besov spaces, Trans. Amer. Math. Soc. 305 (1998) 297-314.
9. D. Donoho, CART and best-ortho-basis: a connection, Ann. Statist. 25 (1997) 1870-1911.
10. C. Fefferman, E. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971) 107-115.
11. Y. Hu, K. Kopotun, X. Yu, On multivariate adaptive approximation, Constr. Approx. 16 (2000) 449-474.
12. G. Kyriazis, P. Petrushev, New bases for Triebel-Lizorkin and Besov spaces, Trans. Amer. Math. Soc. 354 (2002) 749-776.
13. D. Newman, Rational approximation to x , Michigan Math. J. 11 (1964) 11-14.
14. P. Oswald, Multilevel Finite Element Approximation: Theory and Applications, Teubner Skripten zur Numerik, Teubner, Stuttgart, 1994.
15. A. Pekarskii, Relations between best rational and piecewise polynomial approximations, Vestsi Acad. Navuk. BSSR Ser. Fiz.-Mat. Navuk (1986) No. 5, 36-39 (in Russian).
16. p[- 1, 1], Mat. Zametki 39 (1986) 388-394 (English translation in Math. Notes, 39 (1986), 212 216).
17. p metric, J. Approx. Theory 50 (1987) 141-159.
18. P. Petrushev, Direct and converse theorems for spline and rational approximation and Besov spaces, in: M. Cwikel, et al., (Eds.), Function Spaces and Applications, Lecture Notes in Mathematics, Vol. 1302, Springer, Berlin, 1988, pp. 363-377.
19. P. Petrushev, Bases consisting of rational functions of uniformly bounded degrees or more general functions, J. Funct. Anal. 174 (2000) 18-75.
20. P. Petrushev, V. Popov, Rational Approximation of Real Functions, Cambridge University Press, Cambridge, 1987.
21. E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.
22. k), 0
23. V. Temlyakov, The best m-term approximation and greedy algorithms, Adv. Comput. Math. 8 (1998) 249-265.
24. P. Wojtaszczyk, Banach Spaces for Analysts, in: Cambridge Studies in Advanced Mathematics, Vol. 25, Cambridge University Press, Cambridge, 1991.