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On Orthogonal Systems with Extremely Large \(L_2\)-Norm of the Maximal Operator

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The problem of constructing examples that establish the sharpness of the Menchoff– Rademacher theorem on the Weyl multiplier for the almost everywhere convergence of series in general orthogonal systems is considered. We construct an example of a discrete orthonormal system based on \(4\times 4\) blocks such that the partial sums of the series in this system has majorants whose \(L_2\)-norm increases as \(\log_2N\). This orthonormal system is generated by an orthogonal matrix that has improved characteristics, as compared to the Hilbert matrix. We continue the study of B. S. Kashin, who constructed an example: of an orthonormal system, based on binary blocks, with majorant of partial sums increasing as \(\sqrt{\log_2N}\).

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  1. D. Menchoff, “Sur les séries de fonctions orthogonales,” Fund. Math. 4, 82–105 (1923).

    Article  Google Scholar 

  2. H. Rademacher, “Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen,” Math. Ann. 87 (1-2), 112–138 (1922).

    Article  MathSciNet  Google Scholar 

  3. B. S. Kashin and A. A. Saakyan, Orthogonal Series (Izd. AFTs, Moscow, 1999) [in Russian].

    MATH  Google Scholar 

  4. V. I. Matsaev, “A class of completely continuous operators,” Dokl. Akad. Nauk SSSR 139 (3), 548–551 (1961).

    MathSciNet  MATH  Google Scholar 

  5. L. N. Nikol’skaya and J. B. Farforovskaja, “Toeplitz and Hankel matrices as Hadamard–Schur multipliers,” St. Petersburg Math. J. 15 (6), 915–928 (2004).

    Article  MathSciNet  Google Scholar 

  6. A. Paszkiewicz, “A new proof of the Rademacher–Menchoff theorem,” Acta Sci. Math. (Szeged) 71, 631–642 (2005).

    MathSciNet  MATH  Google Scholar 

  7. A. Paszkiewicz, “A complete characterization of coefficients of a.e. convergent orthogonal series and majorizing measures,” Invent. Math. 180 (1), 55–110 (2010).

    Article  MathSciNet  Google Scholar 

  8. A. P. Solodov, “Concerning an example of Paszkiewicz,” Math. Notes 78 (2), 258–263 (2005).

    Article  MathSciNet  Google Scholar 

  9. B. S. Kashin, “Dyadic analogues of Hilbert matrices,” Russian Math. Surveys 71 (6), 1135–1136 (2016).

    Article  MathSciNet  Google Scholar 

  10. E. M. Dyuzhev, “Estimate of the norms of matrices whose entries are constant in binary blocks,” Math. Notes 104 (5), 749–752 (2018).

    Article  MathSciNet  Google Scholar 

  11. E. M. Dyuzhev, “Estimation of norms of matrices with arbitrary elements being constant in binary blocks,” Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., No. 3, 46–48 (2020).

    Google Scholar 

  12. G. A. Karagulyan, “On Weyl multipliers of the rearranged trigonometric system,” Mat. Sb. 211 (12), 49–82 (2020).

    Article  MathSciNet  Google Scholar 

  13. M. A. Naimark, Normalized Rings (GITTLE, Moscow, 1956) [in Russian].

    MATH  Google Scholar 

  14. G. N. Agaev, N. Ya. Vilenkin, G. M. Dzhafarli and A. I. Rubinshtein, Multiplicative Systems of Functions and Harmonic Analysis on Zero-Dimensional Groups (Izd. Élm, Baku, 1981) [in Russian].

    MATH  Google Scholar 

  15. P. L. Ul’yanov, “Weyl factors for unconditional convergence,” Mat. Sb. (N. S.) 60 (102) (1), 39–62 (1963).

    MathSciNet  Google Scholar 

  16. S. V. Bochkarev, “Rearrangements of Fourier–Walsh series,” Math. USSR-Izv. 15 (2), 259–275 (1980).

    Article  Google Scholar 

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Correspondence to A. P. Solodov.

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Translated from Matematicheskie Zametki, 2021, Vol. 109, pp. 436-451

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Solodov, A.P. On Orthogonal Systems with Extremely Large \(L_2\)-Norm of the Maximal Operator. Math Notes 109, 459–472 (2021).

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