Sidonicity and variants of Kaczmarz’s problem. (Sidonicité et un problème de Kaczmarz.) (English. French summary) Zbl 1388.43008
It is well-known, as shown by Rudin, that every Sidon set \(\Lambda\) in a discrete Abelian group is a \(\Lambda (p)\)-set for all \(p < \infty\) and moreover, for \(2 \leq p < \infty\), we have \(\| f \|_p \leq C \, {\sqrt p} \, \|f \|_2\) for every trigonometric polynomial \(f\) with spectrum in \(\Lambda\). G. Pisier [ Ec. Polytech., Cent. Math., 1977-Exposes No. 12, 13, 33 p. (1978; Zbl 0388.43009)] showed spectacularly, using Gaussian processes (and Rider’s theorem), that the converse holds. Other proofs are given by the first author [Ann. Inst. Fourier 35, No. 1, 137–148 (1985; Zbl 0578.43008)] and G. Pisier [Proc. Semin., Torino and Milano 1982, Vol. II, 911–944 (1983; Zbl 0539.43004)], for example.
In the paper under review, the authors study this problem when the characters of the compact Abelian group are replaced by any uniformly bounded orthonormal system (OS in short). It appears that this problem was raised by S. Kaczmarz in the Scottish book, as Problem 130. If \((\phi_j)\) is an OS of complex-valued functions on a probability space \(\Omega\), this system is said to be Sidon if, for all complex numbers: \(\big\| \sum_j a_j \phi_j (\omega) \big\|_\infty \geq c \sum_j |a_j|\), and it is said to satisfy the \(\psi_2\)-condition if \(\big\| \sum_j a_j \phi_j \big\|_p \leq C \, \sqrt{p} \big( \sum_j |a_j|^2\big)^{1/2}\) for all \(p \geq 2\), condition that is equivalent to \(\big\| \sum_j a_j \phi_j \big\|_{\psi_2} \leq C \big( \sum_j |a_j|^2\big)^{1/2}\), where \(\| \, . \, \|_{\psi_2}\) is the norm of the Orlicz space associated to the function \(\psi_2 (x) = \text{e}^{x^2} - 1\).
First of all, the authors show that this \(\psi_2\)-condition does not imply Sidonicity (Theorem 1.2). However, they prove (Theorem 1.3) that this \(\psi_2\)-condition for a uniformly bounded OS \((\phi_j)\) implies that it is a \(5\)-tensor Sidon system, i.e. the OS \((\Phi_j)\) is Sidon, with \(\Phi_j = \phi_j \otimes \phi_j \otimes \phi_j \otimes \phi_j \otimes \phi_j\).
A Rademacher-Sidon system (or Rider system) is a uniformly bounded OS \((\phi_j)\) such that \({\mathbb E}_\omega \big\| \sum_j r_j (\omega) a_j \phi_j \big\|_\infty \geq c \sum_j |a_j|\), where \((r_j)\) is a Rademacher sequence. Clearly, every Sidon OS is Rademacher-Sidon. D. Rider [Duke Math. J. 42, 759–764 (1975; Zbl 0345.43008)] proved the converse when the OS is a set of characters. The authors prove (Theorem 1.4) that if the \(k\)-fold tensor product of an OS is Rademacher-Sidon, then this OS is itself Rademacher-Sidon and (Theorem 1.5) that every OS with the \(\psi_2\)-condition is Rademacher-Sidon.
To prove Theorem 1.3 and Theorem 1.5, the OS \((\phi_j)\) is approximated by a martingale difference sequence and then Riesz product-type arguments are used. The authors give also a more sophisticated proof of Theorem 1.5, using the theory of stochastic processes (Preston’s theorem), Talagrand’s majorizing measure theorem [M. Talagrand, Acta Math. 159, No.1–2, 99–149 (1987; Zbl 0712.60044)] and recent estimates on Bernoulli processes due to W. Bednorz and R. Latała [Ann. Math. (2) 180, No. 3, 1167–1203 (2014; Zbl 1309.60053)].
As a consequence of Theorem 1.5 and of the Elton-Pajor theorem, they get that every uniformly bounded OS with the \(\psi_2\)-condition contains a proportional subsystem which is Sidon.
Note, as quoted at the end of the paper, that G. Pisier [Math. Res. Lett. 24, No. 3, 893–932 (2017; Zbl 1385.43001)] showed that every uniformly bounded OS with the \(\psi_2\)-condition is actually a \(2\)-fold tensor Sidon system, and that being Rademacher-Sidon is equivalent to being \(4\)-fold Rademacher-Sidon.
In the paper under review, the authors study this problem when the characters of the compact Abelian group are replaced by any uniformly bounded orthonormal system (OS in short). It appears that this problem was raised by S. Kaczmarz in the Scottish book, as Problem 130. If \((\phi_j)\) is an OS of complex-valued functions on a probability space \(\Omega\), this system is said to be Sidon if, for all complex numbers: \(\big\| \sum_j a_j \phi_j (\omega) \big\|_\infty \geq c \sum_j |a_j|\), and it is said to satisfy the \(\psi_2\)-condition if \(\big\| \sum_j a_j \phi_j \big\|_p \leq C \, \sqrt{p} \big( \sum_j |a_j|^2\big)^{1/2}\) for all \(p \geq 2\), condition that is equivalent to \(\big\| \sum_j a_j \phi_j \big\|_{\psi_2} \leq C \big( \sum_j |a_j|^2\big)^{1/2}\), where \(\| \, . \, \|_{\psi_2}\) is the norm of the Orlicz space associated to the function \(\psi_2 (x) = \text{e}^{x^2} - 1\).
First of all, the authors show that this \(\psi_2\)-condition does not imply Sidonicity (Theorem 1.2). However, they prove (Theorem 1.3) that this \(\psi_2\)-condition for a uniformly bounded OS \((\phi_j)\) implies that it is a \(5\)-tensor Sidon system, i.e. the OS \((\Phi_j)\) is Sidon, with \(\Phi_j = \phi_j \otimes \phi_j \otimes \phi_j \otimes \phi_j \otimes \phi_j\).
A Rademacher-Sidon system (or Rider system) is a uniformly bounded OS \((\phi_j)\) such that \({\mathbb E}_\omega \big\| \sum_j r_j (\omega) a_j \phi_j \big\|_\infty \geq c \sum_j |a_j|\), where \((r_j)\) is a Rademacher sequence. Clearly, every Sidon OS is Rademacher-Sidon. D. Rider [Duke Math. J. 42, 759–764 (1975; Zbl 0345.43008)] proved the converse when the OS is a set of characters. The authors prove (Theorem 1.4) that if the \(k\)-fold tensor product of an OS is Rademacher-Sidon, then this OS is itself Rademacher-Sidon and (Theorem 1.5) that every OS with the \(\psi_2\)-condition is Rademacher-Sidon.
To prove Theorem 1.3 and Theorem 1.5, the OS \((\phi_j)\) is approximated by a martingale difference sequence and then Riesz product-type arguments are used. The authors give also a more sophisticated proof of Theorem 1.5, using the theory of stochastic processes (Preston’s theorem), Talagrand’s majorizing measure theorem [M. Talagrand, Acta Math. 159, No.1–2, 99–149 (1987; Zbl 0712.60044)] and recent estimates on Bernoulli processes due to W. Bednorz and R. Latała [Ann. Math. (2) 180, No. 3, 1167–1203 (2014; Zbl 1309.60053)].
As a consequence of Theorem 1.5 and of the Elton-Pajor theorem, they get that every uniformly bounded OS with the \(\psi_2\)-condition contains a proportional subsystem which is Sidon.
Note, as quoted at the end of the paper, that G. Pisier [Math. Res. Lett. 24, No. 3, 893–932 (2017; Zbl 1385.43001)] showed that every uniformly bounded OS with the \(\psi_2\)-condition is actually a \(2\)-fold tensor Sidon system, and that being Rademacher-Sidon is equivalent to being \(4\)-fold Rademacher-Sidon.
Reviewer: Daniel Li (Lens)
MSC:
| 43A46 | Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.) |
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
Citations:
Zbl 0388.43009; Zbl 0578.43008; Zbl 0539.43004; Zbl 0345.43008; Zbl 0712.60044; Zbl 1309.60053; Zbl 1385.43001References:
| [1] | Bednorz, Witold; Latała, Rafał, On the boundedness of Bernoulli processes, Ann. Math., 180, 3, 1167-1203 (2014) · Zbl 1309.60053 · doi:10.4007/annals.2014.180.3.8 |
| [2] | Bourgain, Jean, Sidon sets and Riesz products, Ann. Inst. Fourier, 15, 1, 137-148 (1985) · Zbl 0578.43008 · doi:10.5802/aif.1003 |
| [3] | Bourgain, Jean, Geometric aspects of functional analysis, Proc. Isr. Semin., GAFA, Isr. (1989-90), 1469, On the distribution of polynomials on high-dimensional convex sets, 127-137 (1991), Springer · Zbl 0773.46013 |
| [4] | Elton, John, Sign-embeddings of \(l_n^1\), Trans. Am. Math. Soc., 279, 1, 113-124 (1983) · Zbl 0525.46011 |
| [5] | Graham, Colin C.; Hare, Kathryn E., Interpolation and Sidon sets for compact groups, xvii+249 p. pp. (2013), Springer · Zbl 1267.43001 |
| [6] | Lewko, Mark Allison; Lewko, Orthonormal systems in linear spans, Anal. PDE, 7, 1, 97-115 (2014) · Zbl 1298.42032 · doi:10.2140/apde.2014.7.97 |
| [7] | Lopez, Jorge M.; Ross, Kenneth A., Sidon Sets, 13, v+193 p. pp. (1975), Marcel Dekker · Zbl 0351.43008 |
| [8] | Marcus, Michael B.; Pisier, Gilles, Random Fourier series with applications to harmonic analysis, 101, v+150 p. pp. (1981), Princeton University Press and University of Tokyo Press · Zbl 0474.43004 |
| [9] | Mauldin, R. Daniel, The Scottish Book. Mathematics from the Scottish Café, xiii+268 p. pp. (1981), Birkhäuser · Zbl 0485.01013 |
| [10] | Pajor, Alain, Séminaire de géométrie des espaces de Banach, Paris VII, 1983. Tomes I, II, 18, Plongement de \(l_k^1\) complexe dans les espaces de Banach, 139-148 (1984), Université Paris VII · Zbl 0596.46014 |
| [11] | Pisier, Gilles, On uniformly bounded orthonormal Sidon systems · Zbl 1385.43001 |
| [12] | Pisier, Gilles, Ensembles de Sidon et processus gaussiens, C. R. Acad. Sci., Paris, Sér. A, 286, 671-674 (1978) · Zbl 0374.43003 |
| [13] | Preston, Christopher, Banach spaces arising from some integral inequalities, Math. J., Indiana Univ., 20, 997-1015 (1971) · Zbl 0228.46024 · doi:10.1512/iumj.1971.20.20095 |
| [14] | Rider, Daniel, Randomly continuous functions and Sidon sets, Duke Math. J., 42, 4, 759-764 (1975) · Zbl 0345.43008 · doi:10.1215/S0012-7094-75-04264-7 |
| [15] | Rudin, Walter, Some theorems on Fourier coefficients, Proc. Am. Math. Soc., 10, 855-859 (1959) · Zbl 0091.05706 · doi:10.1090/S0002-9939-1959-0116184-5 |
| [16] | Rudin, Walter, Trigonometric series with gaps, J. Math. Mech., 9, 203-227 (1960) · Zbl 0091.05802 |
| [17] | Spencer, Joel, Six standard deviations suffice, Trans. Am. Math. Soc., 289, 2, 679-706 (1985) · Zbl 0577.05018 · doi:10.1090/S0002-9947-1985-0784009-0 |
| [18] | Talagrand, Michel, Regularity of Gaussian processes, Acta Math., 159, 1-2, 99-149 (1987) · Zbl 0712.60044 · doi:10.1007/BF02392556 |
| [19] | Talagrand, Michel, Majorizing measures without measures, Ann. Probab., 29, 1, 411-417 (2001) · Zbl 1019.60033 · doi:10.1214/aop/1008956336 |
| [20] | Talagrand, Michel, Upper and lower bounds for stochastic processes. Modern methods and classical problems, 60, xv+626 p. pp. (2014), Springer · Zbl 1293.60001 |
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