Ganichev, M.; Kalton, N. J. Convergence of the weak dual greedy algorithm in \(L_{p}\)-spaces. (English) Zbl 1041.46007 J. Approximation Theory 124, No. 1, 89-95 (2003). A dictionary in a real Banach space \(X\) is a subset \(D\subseteq X\) which is (i) normalized: \(g\in D\) implies \(\| g\|=1\); (ii) symmetric: \(-D=D\); (iii) fundamental: \(X\) is the closed linear space generated by \(D\). The goal of the present paper is to construct algorithms giving, for each \(x\in X\), a sequence \(\sum_{k=1}^n t_k g_k\), with non-negative real numbers \(t_k\) and \(g_k\in D\), and which (as expected) converges to \(x\).The pure greedy algorithm (PGA) was introduced and studied by P. Huber [Ann. Stat. 13, 435–525 (1985; Zbl 0595.62059)]; its convergence was proved by L. Jones [Ann. Stat. 15, 880–882 (1987; Zbl 0664.62061)]. In Hilbert spaces, it was modified into the weak greedy algorithm (WGA) and its convergence was proved by L. Rejtö and G. Walter [Stochastic Anal. Appl. 10, No. 2, 213–222 (1992; Zbl 0763.62034)].In the paper under review, the authors study a generalization, the weak dual greedy algorithm (WDGA), in Banach spaces with a Gâteaux differentiable norm. It was conjectured by V. Temlyakov [Found. Comput. Math. 3, No. 1, 33–107 (2003; Zbl 1039.41012)] that (WDGA) converges when \(X\) is a uniformly smooth Banach space with power-type modulus of smoothness. The authors prove the conjecture when \(X\) is a subspace of a quotient of \(L^p\), \(1<p<+\infty\). For that purpose, they define a geometric property, called \(\Gamma\), in Banach spaces with a Gâteaux differentiable norm, and they prove that:(1) every subspace of a quotient of \(L^p\), \(1<p<+\infty\), has property \(\Gamma\) (the authors acknowledge that for \(p\geq 2\), the result was essentially proved by E. D. Livshits [Math. Notes 73, 342–358 (2003)]; the key fact for their proof is the following elementary inequality: \(b| a+b|^{p-1} \text{sgn}(a+b) -b| a|^{p-1} \text{sgn}(a)\leq C_p(| a+b|^p-p\,b| a|^{p-1} \text{sgn}(a)-| a|^p)\);(2) in every Banach space \(X\) with Fréchet differentiable norm and which has property \(\Gamma\), the (WDGA) converges for every \(x\in X\) and every dictionary.This result is new even when \(X\) is \(L^p\) itself and the dictionary \(D\) is the Haar basis. When \(D\) is the trigonometric system, see [S. Dilworth, D. Kutzarova and V. Temlyakov, J. Fourier Anal. Appl. 8, No. 5, 489–505 (2002; Zbl 1025.41021)]. Reviewer: Daniel Li (Lens) Cited in 1 ReviewCited in 9 Documents MSC: 46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces 46B03 Isomorphic theory (including renorming) of Banach spaces 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) Keywords:dictionary; Fréchet differentiable norm; Gâteaux differentiable norm; greedy algorithm Citations:Zbl 0595.62059; Zbl 0664.62061; Zbl 0763.62034; Zbl 1025.41021; Zbl 1039.41012 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] R. Deville, G. Godefroy, V. Zizler, Smoothness and Renormings in Banach Spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 64, Longman Scientific and Technical, Harlow, 1993.; R. Deville, G. Godefroy, V. Zizler, Smoothness and Renormings in Banach Spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 64, Longman Scientific and Technical, Harlow, 1993. · Zbl 0782.46019 [2] Dilworth, S. J.; Kutzarova, D.; Temlyakov, V. N., Convergence of some greedy algorithms in Banach spaces, J. Fourier Anal. Appl., 8, 489-505 (2002) · Zbl 1025.41021 [3] Huber, P. J., Projection pursuit, Ann. Statist., 13, 435-525 (1985) · Zbl 0595.62059 [4] Jones, L. K., On a conjecture of Huber concerning the convergence of projection pursuit regression, Ann. Statist., 15, 880-882 (1987) · Zbl 0664.62061 [5] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces, Vol. II, Function Spaces, Springer, Berlin, 1979.; J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces, Vol. II, Function Spaces, Springer, Berlin, 1979. · Zbl 0403.46022 [6] Livshits, E. D., On convergence of greedy algorithms in Banach spaces, Mat. Zametki, 73, 371-389 (2003) · Zbl 1058.41036 [7] E.D. Livshits, V.N. Temlyakov, On the convergence of a weak greedy algorithm, (Russian) Tr. Mat. Inst. Steklova 232 (2001) 236-247; translation in Proc. Steklov Inst. Math. 232(1) (2001) 229-239.; E.D. Livshits, V.N. Temlyakov, On the convergence of a weak greedy algorithm, (Russian) Tr. Mat. Inst. Steklova 232 (2001) 236-247; translation in Proc. Steklov Inst. Math. 232(1) (2001) 229-239. · Zbl 1003.65011 [8] Rejto, L.; Walter, G. G., Remarks on projection pursuit regression and density estimation, Stochastic Anal. Appl., 10, 213-222 (1992) · Zbl 0763.62034 [9] Temlyakov, V. N., Weak greedy algorithms, Adv. Comput. Math., 12, 213-227 (2000) · Zbl 0964.65009 [10] Temlyakov, V. N., A criterion for convergence of weak greedy algorithms, Adv. Comput. Math., 17, 269-280 (2002) · Zbl 1128.41309 [11] Temlyakov, V. N., Nonlinear methods of approximation, Found. Comput. 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