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The Marcinkiewicz-type discretization theorems. (English) Zbl 1404.41012
Let \(\Omega \) be a compact subset of \(\mathbb{R}^d,\, \mu\) a probability measure on \(\Omega\) and \(1\leq q<\infty.\) One says that an \(N\)-dimensional subspace \(X_N\) of \(L_q(\Omega,\mu)\) satisfies a Marcinkiewicz type theorem if there exist a subset \(\{\xi^\nu: 1\leq\nu\leq m\}\) of \(\Omega\) and two constants \(C_1(d,q),C_2(d,q)>0\) such that
\[ C_1(d,q)\|f\|^q_q\leq \frac1m\sum_{\nu=1}^m|f(\xi^\nu)|^q\leq C_2(d,q)\|f\|^q_q\,,(\ast) \] for all \(f\in X_N.\) In the case of the uniform norm one asks that \[ C_1(d,q)\|f\|_\infty\leq\max_{1\leq \nu\leq m}|f(\xi^\nu)| \leq\|f\|_\infty\,, \] for all \(f\in X_N\subset C(\Omega).\)
As the author mention, such a result was obtained for the first time by J. Marcinkiewicz in the case of univariate trigonometric polynomials:
\[ C_1(q)\|f\|^q_q\leq \frac1m\sum_{\nu=1}^{2n+1}|f\big(2\pi\nu/(2n+1)\big)|^q\leq C_2(q)\|f\|^q_q\,. \] The author considers also some variants of Marcinkiewicz problem:
\(\bullet\) the Marcinkiewicz problem with weights (with \(\sum_{\nu=1}^m\lambda_\nu|f(\xi^\nu)|^q \) in \((*)\));
\(\bullet\) the Marcinkiewicz problem with \(\varepsilon\) (when \(C_1(d,q)=1-\varepsilon\) and \(C_2(d,q)=1+\varepsilon\)).
The aim of the paper is “to present here a new technique, which works well for discretization of the integral norm. It is a combination of probabilistic technique, based on chaining, and results on the entropy numbers in the uniform norm.” (quoted from abstract).

MSC:
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
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References:
[1] Batson, J; Spielman, DA; Srivastava, N, Twice-Ramanujan sparsifiers, SIAM J. Comput., 41, 1704-1721, (2012) · Zbl 1260.05092
[2] Bourgain, J; Lindenstrauss, J; Milman, V, Approximation of zonoids by zonotopes, Acta Math., 162, 73-141, (1989) · Zbl 0682.46008
[3] Dũng, D., Temlyakov, V.N., Ullrich, T.: Hyperbolic cross approximation (2016). arXiv:1601.03978v2 [math.NA]
[4] Donahue, M; Gurvits, L; Darken, C; Sontag, E, Rate of convex approximation in non-Hilbert spaces, Constr. Approx., 13, 187-220, (1997) · Zbl 0876.41016
[5] Kashin, BS, Lunin’s method for selecting large submatrices with small norm, Matem. Sb., 206, 95-102, (2015) · Zbl 1336.46014
[6] Kashin, BS; Temlyakov, VN, On a norm and related applications, Mat. Zametki, 64, 637-640, (1998) · Zbl 0955.42002
[7] Kashin, B.S., Temlyakov, V.N.: On a norm and approximation characteristics of classes of functions of several variables. Metric theory of functions and related problems in analysis, Izd. Nauchno-Issled. Aktuarno-Finans. Tsentra (AFTs), Moscow, pp. 69-99 (1999)
[8] Kashin, BS; Temlyakov, VN, The volume estimates and their applications, East J. Approx., 9, 469-485, (2003) · Zbl 1111.41019
[9] Marcus, A; Spielman, DA; Srivastava, N, Interlacing families II: mixed characteristic polynomials and the kadison-Singer problem, Ann. Math., 182, 327-350, (2015) · Zbl 1332.46056
[10] Nitzan, S; Olevskii, A; Ulanovskii, A, Exponential frames on unbounded sets, Proc. Am. Math. Soc., 144, 109-118, (2016) · Zbl 1327.42035
[11] Rudelson, M, Almost orthogonal submatrices of an orthogonal matrix, Izr. J. Math., 111, 143-155, (1999) · Zbl 0958.15019
[12] Temlyakov, VN, Weak greedy algorithms, Adv. Comput. Math., 12, 213-227, (2000) · Zbl 0964.65009
[13] Temlyakov, VN, Greedy algorithms in Banach spaces, Adv. Comput. Math., 14, 277-292, (2001) · Zbl 0988.41022
[14] Temlyakov, VN, Greedy-type approximation in Banach spaces and applications, Constr. Approx., 21, 257-292, (2005) · Zbl 1070.41018
[15] Temlyakov, V.N.: Greedy Approximation. Cambridge University Press, Cambridge (2011) · Zbl 1279.41001
[16] Temlyakov, VN, An inequality for the entropy numbers and its application, J. Approx. Theory, 173, 110-121, (2013) · Zbl 1283.41021
[17] Temlyakov, V.N.: Incremental greedy algorithm and its applications in numerical integration. In: Springer Proceedings in Mathematics and Statistics, Monte Carlo and Quasi-Monte Carlo Methods, MCQMC, Leuven, Belgium, Apr 2014, pp. 557-570 (2014) · Zbl 1356.65088
[18] Temlyakov, V.N.: Constructive sparse trigonometric approximation and other problems for functions with mixed smoothness. Matem. Sb. 206, 131-160 (2015). arXiv:1412.8647v1 [math.NA] · Zbl 1362.41009
[19] Temlyakov, V.N.: Constructive sparse trigonometric approximation for functions with small mixed smoothness. Constr. Approx. 45, 467-495 (2017). arXiv:1503.0282v1 [math.NA] · Zbl 1373.42009
[20] Temlyakov, V.N.: On the entropy numbers of the mixed smoothness function classes. J. Approx. Theory 207, 26-56 (2017). arXiv:1602.08712v1 [math.NA] · Zbl 1366.41019
[21] Temlyakov, V.N.: The Marcinkewiecz-type discretization theorems for the hyperbolic cross polynomials. Jaen J. Approx. 9(1), 37-63 (2017). arXiv:1702.01617v2 [math.NA]
[22] Tropp, JA, User-friendly tail bounds for sums of random matrices, Found. Comput. Math., 12, 389-434, (2012) · Zbl 1259.60008
[23] Zygmund, A.: Trigonometric Series. Cambridge University Press, Cambridge (1959) · Zbl 0085.05601
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