Geodesic
Non-linear approximation of continuous functions
Sbornik. Mathematics, Tome 199 (2008) no. 5, pp. 629-653 Cet article a éte moissonné depuis la source Math-Net.Ru

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The existence of a function $f_0(x)\in C_{[0,1]}$ for which the greedy algorithm in the Faber-Schauder system is divergent in measure on $[0,1]$ is established. It is shown that for each $\varepsilon$, $0<\varepsilon<1$, there exists a measurable subset $E$ of $ [0,1]$ of measure $|E|>1-\varepsilon$ such that for each $f(x)\in C_{[0,1]}$ one can find a function $\widetilde f(x)\in C_{[0,1]}$ coinciding with $f(x)$ on $E$, whose greedy algorithm in the Faber-Schauder system converges uniformly on $[0,1]$. Bibliography: 33 titles. 
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M. G. Grigoryan; A. A. Sargsyan. Non-linear approximation of continuous functions. Sbornik. Mathematics, Tome 199 (2008) no. 5, pp. 629-653. http://geodesic.mathdoc.fr/item/SM_2008_199_5_a0/

[1] N. N. Luzin, “K osnovnoi teoreme integralnogo ischisleniya”, Matem. sb., 28:2 (1912), 266–294 | Zbl

[2] D. Menchoff, “Sur la convergence uniforme des séries de Fourier”, Matem. sb., 11(53):1–2 (1942), 67–96 | MR | Zbl

[3] A. A. Talalyan, “Dependence of convergence of orthogonal series on changes of values of the function expanded”, Math. Notes, 33:5 (1983), 368–372 | DOI | MR | Zbl

[4] O. D. Tsereteli, “O skhodimosti pochti vsyudu ryadov Fure”, Soobsch. AN GSSR, 57 (1970), 21–24 | MR | Zbl

[5] J. J. Price, “Walsh series and adjustment of functions on small sets”, Illinois J. Math., 13 (1969), 131–136 | MR | Zbl

[6] K. I. Oskolkov, “Uniform modulus of continuity of summable functions on sets of positive measure”, Soviet Math. Dokl., 17:2 (1976), 1028–1030 | MR | Zbl

[7] B. S. Kashin, G. G. Kosheleva, “An approach to “correction” theorems”, Moscow Univ. Math. Bull., 43:4 (1988), 1–5 | MR | Zbl

[8] Š. V. Heladze, “Convergence of Fourier series almost everywhere and in the $L$-metric”, Math. USSR-Sb., 35:4 (1979), 527–539 | DOI | MR | Zbl | Zbl

[9] M. G. Grigorian, K. S. Kazarian, F. Soria, “Mean convergence of orthogonal Fourier series of modified functions”, Trans. Amer. Math. Soc., 352:8 (2000), 3777–3798 | DOI | MR | Zbl

[10] M. G. Grigorian, “On the convergence of Fourier series in the metric of $L^1$”, Anal. Math., 17:3 (1991), 211–237 | DOI | MR | Zbl

[11] M. G. Grigoryan, “On convergence of Fourier series in complete orthonormal systems in the $L^1$-metric and almost everywhere”, Math. USSR-Sb., 70:2 (1991), 445–466 | DOI | MR | Zbl | Zbl

[12] M. G. Grigorian, “On the $L^p_\mu$-strong property of orthonormal systems”, Sb. Math., 194:10 (2003), 1503–1532 | DOI | MR | Zbl

[13] M. G. Grigoryan, “The almost everywhere convergence of Fourier series according to complete orthonormal systems”, Math. Notes, 51:5 (1992), 447–453 | DOI | MR | Zbl

[14] M. G. Grigorian, “On the representation of functions by orthogonal series in weighted $L^p$ spaces”, Studia Math., 134:3 (1999), 207–216 | MR | Zbl

[15] D. E. Menshov, “O ryadakh Fure nepreryvnykh funktsii”, Uchenye zapiski MGU, 148 (1951), 108–132 | MR

[16] D. E. Menshov, “O ryadakh Fure ot summiruemykh funktsii”, Tr. MMO, 1, 1952, 5–38 | MR | Zbl

[17] V. N. Temlyakov, “Nonlinear methods of approximation”, Found. Comput. Math., 3:1 (2003), 33–107 | DOI | MR | Zbl

[18] R. A. DeVore, V. N. Temlyakov, “Some remarks on greedy algorithms”, Adv. Comput. Math., 5:1 (1996), 173–187 | DOI | MR | Zbl

[19] S. V. Konyagin, V. N. Temlyakov, “A remark on greedy approximation in Banach spaces”, East J. Approx., 5:3 (1999), 365–379 | MR | Zbl

[20] P. Wojtaszczyk, “Greedy algorithm for general biorthogonal systems”, J. Approx. Theory, 107:2 (2000), 293–314 | DOI | MR | Zbl

[21] T. W. Körner, “Divergence of decreasing rearranged Fourier series”, Ann. of Math. (2), 144:1 (1996), 167–180 | DOI | MR | Zbl

[22] M. G. Grigoryan, “On the convergence in the $L^p$-metric of a greedy algorithm in a trigonometric series”, J. Contemp. Math. Anal., 39:5 (2005), 35–48 | MR

[23] M. G. Grigoryan, S. L. Gogyan, “Nelineinaya approksimatsiya po sisteme Khaara i modifikatsii funktsii”, Anal. Math., 32:1 (2006), 49–80 | DOI | MR | Zbl

[24] S. Gogyan, “Greedy algorithm with regard to Haar subsystems”, East J. Approx., 11:2 (2005), 221–236 | MR

[25] S. J. Dilworth, N. J. Kalton, D. Kutzarova, “On the existence of almost greedy bases in Banach spaces”, Studia Math., 159:1 (2003), 67–101 | DOI | MR | Zbl

[26] A. A. Sargsyan, “O kvazigridibazisnosti sistemy Fabera–Shaudera”, Dokl. NAN Armenii, 105:4 (2005), 333–337 | MR

[27] A. A. Sargsyan, “The quasi-greedy system property of some subsystems of the Faber–Schauder system in $C(0,1)$”, J. Contemp. Math. Anal., 40:3 (2005), 44–53 | MR

[28] A. A. Sargsyan, “On the quasi-greedy system and democracy properties of some subsystems of the Faber–Schauder system”, J. Contemp. Math. Anal., 41:2 (2007), 52–69 | MR

[29] A. V. Sil'nichenko, “Rate of convergence of greedy algorithms”, Math. Notes, 76:3–4 (2004), 582–586 | DOI | MR | Zbl

[30] E. D. Livshits, “Optimality of the greedy algorithm for some function classes”, Sb. Math., 198:5 (2007), 691–709 | DOI | MR

[31] J. Schauder, “Zur Theorie stetiger Abbildungen in Funktionalräumen”, Math. Z., 26:1 (1927), 47–65 | DOI | MR | Zbl

[32] P. L. Ul'yanov, “Representation of functions by series and classes $\varphi(L)$”, Russian Math. Surveys, 27:2 (1972), 1–54 | DOI | MR | Zbl

[33] G. M. Fikhtengol'ts, Differential- und Integralrechnung, II, Hochschulbucher fur Math., 62, VEB Deutscher Verlag, Berlin, 1986 | MR | Zbl | Zbl