Geodesic
Dense weakly lacunary subsystems of orthogonal systems and maximal partial sum operator
Sbornik. Mathematics, Tome 214 (2023) no. 11, pp. 1560-1584

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It is shown that any finite orthogonal system of functions whose norms in $L_p$ are bounded by 1, where $p>2$, has a sufficiently dense subsystem with lacunarity property in the Orlicz space. The norm of the maximal partial sum operator for this subsystem has a better estimate than it is guaranteed by the classical Menshov-Rademacher theorem for general orthogonal systems. Bibliography: 17 titles.
Keywords: lacunary subsystems, maximal partial sum operator, Orlicz space 
I. V. Limonova. Dense weakly lacunary subsystems of orthogonal systems and maximal partial sum operator. Sbornik. Mathematics, Tome 214 (2023) no. 11, pp. 1560-1584. http://geodesic.mathdoc.fr/item/SM_2023_214_11_a2/
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[1] I. Agaev, “Lacunary subsets of orthonormal sets”, Anal. Math., 11:4 (1985), 283–301 | DOI | MR | Zbl

[2] T. O. Balykbaev, “On a class of lacunary orthonormal systems”, Soviet Math. Dokl., 33 (1986), 267–269 | MR | Zbl

[3] T. O. Balykbaev, A class of lacunary trigonometric systems, Kandidat Dissertation, Moscow State University, Moscow, 1986, 67 pp. (Russian)

[4] S. Banach, “Sur les séries lacunaires”, Bull. Int. Acad. Pol. Sci. Lett. Cl. Sci. Math. Nat. Ser. A Sci. Math., 1933 (1933), 149–154 | Zbl

[5] J. Bourgain, “Bounded orthogonal systems and the $\Lambda(p)$-set problem”, Acta Math., 162:3–4 (1989), 227–245 | DOI | MR | Zbl

[6] J. Bourgain, “On Kolmogorov's rearrangement problem for orthogonal systems and Garsia's conjecture”, Geometric aspects of functional analysis, Israel seminar (GAFA) (1987–88), Lecture Notes in Math., 1376, Springer-Verlag, Berlin, 1989, 209–250 | DOI | MR | Zbl

[7] V. F. Gaposhkin, “Lacunary series and independent functions”, Russian Math. Surveys, 21:6 (1966), 1–82 | DOI | MR | Zbl

[8] O. Guédon, S. Mendelson, A. Pajor and N. Tomczak-Jaegermann, “Subspaces and orthogonal decompositions generated by bounded orthogonal systems”, Positivity, 11:2 (2007), 269–283 | DOI | MR | Zbl

[9] S. Kaczmarz and H. Steinhaus, Theorie der Orthogonalreihen, Monogr. Mat., 6, Subwencji funduszu kultury narodowej, Warszawa–Lwow, 1935, vi+298 pp. | MR | Zbl

[10] G. A. Karagulyan, “On the selection of a convergence subsystem with logarithmic density from an arbitrary orthonormal system”, Math. USSR-Sb., 64:1 (1989), 41–56 | DOI | MR | Zbl

[11] B. S. Kašin (Kashin), “On unconditional convergence in the space $L_1$”, Math. USSR-Sb., 23:4 (1974), 509–519 | DOI | MR | Zbl

[12] B. S. Kashin and I. V. Limonova, “Selecting a dense weakly lacunary subsystem in a bounded orthonormal system”, Russian Math. Surveys, 74:5 (2019), 956–958 | DOI | MR | Zbl

[13] B. S. Kashin and I. V. Limonova, “Weakly lacunary orthogonal systems and properties of the maximal partial sum operator for subsystems”, Proc. Steklov Inst. Math., 311 (2020), 152–170 | DOI | MR | Zbl

[14] B. S. Kashin and A. A. Saakyan, Orthogonal series, 2nd augmented ed., Actuarial and Foinancial Center, Moscow, 1999, x+550 pp. ; English transl. of 1st ed., Transl. Math. Monogr., 75, Amer. Math. Soc., Providence, RI, 1989, xii+451 pp. | MR | Zbl | DOI | MR | Zbl

[15] I. V. Limonova, “Existence of dense subsystems with lacunarity property in orthogonal systems”, Russian Math. Surveys, 77:5 (2022), 952–954 | DOI | MR | Zbl

[16] I. V. Limonova, Restrictions of operators to coordinate subspaces and discretization theorems, Kandidat Dissertation, Moscow State University, Moscow, 2022, 81 pp. (Russian) https://www.mi-ras.ru/dis/ref22/limonova/dis.pdf

[17] M. Talagrand, “Sections of smooth convex bodies via majorizing measures”, Acta Math., 175:2 (1995), 273–300 | DOI | MR | Zbl