Nonlinear $N$-term approximation by refinable functions
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Yu. Brudnyĭ
Translated by: the author - St. Petersburg Math. J. 16 (2005), 143-179
- DOI: https://doi.org/10.1090/S1061-0022-04-00846-5
- Published electronically: December 14, 2004
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Abstract:
Several almost optimal results are obtained about $N$-term nonlinear approximation by dilated integer translates of a refinable function associated with a finite mask and a rather general matrix dilation $A\in GL_n({\mathbb {Z}})$.References
- Christoph Bandt, Self-similar sets. V. Integer matrices and fractal tilings of $\textbf {R}^n$, Proc. Amer. Math. Soc. 112 (1991), no. 2, 549–562. MR 1036982, DOI 10.1090/S0002-9939-1991-1036982-1
- Ju. A. Brudnyĭ, Rational approximation and imbedding theorems, Dokl. Akad. Nauk SSSR 247 (1979), no. 2, 269–272 (Russian). MR 545347
- Yu. A. Brudnyĭ, Adaptive approximation of functions with singularities, Trudy Moskov. Mat. Obshch. 55 (1994), 149–242, 365 (Russian, with Russian summary); English transl., Trans. Moscow Math. Soc. (1994), 123–186 (1995). MR 1468457
- Yuri Brudnyi, Nonlinear piecewise polynomial approximation of functions from Besov spaces, Math. Balkanica (N.S.) 16 (2002), no. 1-4, 339–358. Dedicated to the 70th anniversary of Academician Blagovest Sendov. MR 1934002
- Yu. A. Brudnyĭ and I. P. Irodova, Nonlinear spline approximation of functions of several variables and B-spaces, Algebra i Analiz 4 (1992), no. 4, 45–79 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 4 (1993), no. 4, 667–694. MR 1190782
- Yu. Brudnyi and Inna Kozlov, An algorithm of nonlinear approximation by piecewise polynomials, Mat. Fiz. Anal. Geom. 9 (2002), no. 3, 311–325. MR 1949788
- Ju. A. Brudnyĭ and N. Ja. Krugljak, A family of approximation spaces, Studies in the theory of functions of several real variables, No. 2 (Russian), Yaroslav. Gos. Univ., Yaroslavl′1978, pp. 15–42 (Russian). MR 559913
- Yu. A. Brudnyĭ and N. Ya. Krugljak, Interpolation functors and interpolation spaces. Vol. I, North-Holland Mathematical Library, vol. 47, North-Holland Publishing Co., Amsterdam, 1991. Translated from the Russian by Natalie Wadhwa; With a preface by Jaak Peetre. MR 1107298
- Ju. A. Brudnyĭ and B. D. Kotljar, A certain problem of combinatorial geometry, Sibirsk. Mat. Ž. 11 (1970), 1171–1173, 1199 (Russian). MR 0270270
- Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275
- M. Š. Birman and M. Z. Solomjak, Piecewise polynomial approximations of functions of classes $W_{p}{}^{\alpha }$, Mat. Sb. (N.S.) 73 (115) (1967), 331–355 (Russian). MR 0217487
- M. Š. Birman and M. Z. Solomjak, Estimates for the singular numbers of integral operators, Uspehi Mat. Nauk 32 (1977), no. 1(193), 17–84, 271 (Russian). MR 0438186
- M. Š. Birman and M. Z. Solomjak, Quantitative analysis in Sobolev’s imbedding theorems and applications to spectral theory, Tenth Mathematical School (Summer School, Kaciveli/Nalchik, 1972) Izdanie Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1974, pp. 5–189 (Russian). MR 0482138
- Alfred S. Cavaretta, Wolfgang Dahmen, and Charles A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc. 93 (1991), no. 453, vi+186. MR 1079033, DOI 10.1090/memo/0453
- A. Cohen, K. Gröchenig, and L. F. Villemoes, Regularity of multivariate refinable functions, Constr. Approx. 15 (1999), no. 2, 241–255. MR 1668921, DOI 10.1007/s003659900106
- Ronald A. DeVore, Nonlinear approximation, Acta numerica, 1998, Acta Numer., vol. 7, Cambridge Univ. Press, Cambridge, 1998, pp. 51–150. MR 1689432, DOI 10.1017/S0962492900002816
- Ronald A. DeVore, Björn Jawerth, and Vasil Popov, Compression of wavelet decompositions, Amer. J. Math. 114 (1992), no. 4, 737–785. MR 1175690, DOI 10.2307/2374796
- R. A. DeVore, P. Petrushev, and X. M. Yu, Nonlinear wavelet approximation in the space $C(\textbf {R}^d)$, Progress in approximation theory (Tampa, FL, 1990) Springer Ser. Comput. Math., vol. 19, Springer, New York, 1992, pp. 261–283. MR 1240786, DOI 10.1007/978-1-4612-2966-7_{1}1
- John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
- X.-G. He, K.-S. Lau, and H. Rao, Self-affine sets and graph-directed systems, Constr. Approx. 19 (2003), 373–397; Preprint, City Univ. of Hong-Kong, 2002, 27 pp.
- R. Q. Jia, Approximation properties of multivariate wavelets, Preprint, Univ. of Alberta, Edmonton, 1996.
- Rong Qing Jia and Charles A. Micchelli, On linear independence for integer translates of a finite number of functions, Proc. Edinburgh Math. Soc. (2) 36 (1993), no. 1, 69–85. MR 1200188, DOI 10.1017/S0013091500005903
- Pierre-Gilles Lemarié-Rieusset, Projecteurs invariants, matrices de dilatation, ondelettes et analyses multi-résolutions, Rev. Mat. Iberoamericana 10 (1994), no. 2, 283–347 (French, with English and French summaries). MR 1286477, DOI 10.4171/RMI/153
- Jeffrey C. Lagarias and Yang Wang, Self-affine tiles in $\textbf {R}^n$, Adv. Math. 121 (1996), no. 1, 21–49. MR 1399601, DOI 10.1006/aima.1996.0045
- P. P. Petrushev and V. A. Popov, Rational approximation of real functions, Encyclopedia of Mathematics and its Applications, vol. 28, Cambridge University Press, Cambridge, 1987. MR 940242
- Kenneth H. Rosen, John G. Michaels, Jonathan L. Gross, Jerrold W. Grossman, and Douglas R. Shier (eds.), Handbook of discrete and combinatorial mathematics, CRC Press, Boca Raton, FL, 2000. MR 1725200
- E. Schmidt, Zur Theorie der linearen und nonlinearen Integralgleichungen. I, Math. Ann. 63 (1907), 433–476.
- V. N. Temlyakov, Nonlinear methods of approximation, IMI–Preprint Ser., Univ. of South Caroline, 2001, 1–57.
- P. Wojtaszczyk, A mathematical introduction to wavelets, London Mathematical Society Student Texts, vol. 37, Cambridge University Press, Cambridge, 1997. MR 1436437, DOI 10.1017/CBO9780511623790
- Ding-Xuan Zhou, Self-similar lattice tilings and subdivision schemes, SIAM J. Math. Anal. 33 (2001), no. 1, 1–15. MR 1857987, DOI 10.1137/S0036141000367977
Bibliographic Information
- Yu. Brudnyĭ
- Affiliation: Department of Mathematics, Technion—Israel Institute of Technology, 32000 Haifa, Israel
- Email: ybrudnyi@tx.technion.ac.il
- Received by editor(s): May 15, 2003
- Published electronically: December 14, 2004
- Additional Notes: Supported by the Fund for the Promotion of Research at the Technion.
- © Copyright 2004 American Mathematical Society
- Journal: St. Petersburg Math. J. 16 (2005), 143-179
- MSC (2000): Primary 41A63, 41A30
- DOI: https://doi.org/10.1090/S1061-0022-04-00846-5
- MathSciNet review: 2068353
Dedicated: Dedicated to my friend Misha Birman with love and gratitude