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Remarks on Sampling Discretization of Integral Norms of Functions

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Abstract

We complement a result due to I. V. Limonova and V. N. Temlyakov on sampling discretization of the \(L^2\)-norm and make some general comments concerning the problem of sampling discretization.

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References

  1. J. Batson, D. A. Spielman, and N. Srivastava, “Twice-Ramanujan sparsifiers,” SIAM J. Comput. 41 (6), 1704–1721 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Bourgain, J. Lindenstrauss, and V. Milman, “Approximation of zonoids by zonotopes,” Acta Math. 162 (1–2), 73–141 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  3. F. Dai, A. Prymak, A. Shadrin, V. Temlyakov, and S. Tikhonov, “Sampling discretization of integral norms,” Constr. Approx. 54 (3), 455–471 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  4. F. Dai, A. Prymak, A. Shadrin, V. Temlyakov, and S. Tikhonov, “Entropy numbers and Marcinkiewicz-type discretization,” J. Funct. Anal. 281 (6), 109090 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  5. F. Dai, A. Prymak, V. N. Temlyakov, and S. Yu. Tikhonov, “Integral norm discretization and related problems,” Russ. Math. Surv. 74 (4), 579–630 (2019) [transl. from Usp. Mat. Nauk 74 (4), 3–58 (2019)].

    Article  MathSciNet  MATH  Google Scholar 

  6. F. Dai and V. Temlyakov, “Universal sampling discretization,” arXiv: 2107.11476 [math.FA].

  7. F. Dai and V. Temlyakov, “Sampling discretization of integral norms and its application,” arXiv: 2109.09030 [math.NA].

  8. N. J. A. Harvey and N. Olver, “Pipage rounding, pessimistic estimators and matrix concentration,” in Proc. 25th Annu. ACM–SIAM Symp. on Discrete Algorithms, SODA 2014 (SIAM, Philadelphia, PA, 2014), pp. 926–945.

    Google Scholar 

  9. B. S. Kashin, “Lunin’s method for selecting large submatrices with small norm,” Sb. Math. 206 (7), 980–987 (2015) [transl. from Mat. Sb. 206 (7), 95–102 (2015)].

    Article  MathSciNet  MATH  Google Scholar 

  10. B. S. Kashin, “Decomposing an orthogonal matrix into two submatrices with extremally small \((2,1)\)-norm,” Russ. Math. Surv. 72 (5), 971–973 (2017) [transl. from Usp. Mat. Nauk 72 (5), 193–194 (2017)].

    Article  MathSciNet  MATH  Google Scholar 

  11. B. Kashin, S. Konyagin, and V. Temlyakov, “Sampling discretization of the uniform norm,” arXiv: 2104.01229 [math.NA].

  12. B. Kashin, E. Kosov, I. Limonova, and V. Temlyakov, “Sampling discretization and related problems,” J. Complexity 71, 101653 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  13. B. S. Kashin and I. V. Limonova, “Decomposing a matrix into two submatrices with extremally small \((2,1)\)-norm,” Math. Notes 106 (1–2), 63–70 (2019) [transl. from Mat. Zametki 106 (1), 53–61 (2019)].

    Article  MathSciNet  MATH  Google Scholar 

  14. B. S. Kashin and V. N. Temlyakov, “Observations on discretization of trigonometric polynomials with given spectrum,” Russ. Math. Surv. 73 (6), 1128–1130 (2018) [transl. from Usp. Mat. Nauk 73 (6), 197–198 (2018)].

    Article  MATH  Google Scholar 

  15. E. D. Kosov, “Marcinkiewicz-type discretization of \(L^p\)-norms under the Nikolskii-type inequality assumption,” J. Math. Anal. Appl. 504 (1), 125358 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Ledoux and M. Talagrand, Probability in Banach Spaces: Isoperimetry and Processes (Springer, Berlin, 2013).

    MATH  Google Scholar 

  17. I. V. Limonova, “Decomposing a matrix into two submatrices with smaller \((2,1)\)-norms,” Russ. Math. Surv. 71 (4), 781–783 (2016) [transl. from Usp. Mat. Nauk 71 (4), 185–186 (2016)].

    Article  MathSciNet  MATH  Google Scholar 

  18. I. V. Limonova, “Decomposing a matrix into two submatrices with extremely small operator norm,” Math. Notes 108 (1–2), 137–141 (2020) [transl. from Mat. Zametki 108 (1), 153–157 (2020)].

    Article  MathSciNet  MATH  Google Scholar 

  19. I. V. Limonova, “Exact discretization of the \(L_2\)-norm with negative weight,” Math. Notes 110 (3–4), 458–462 (2021) [transl. from Mat. Zametki 110 (3), 465–470 (2021)].

    Article  MathSciNet  MATH  Google Scholar 

  20. I. Limonova and V. Temlyakov, “On sampling discretization in \(L_2\),” J. Math. Anal. Appl. 515 (2), 126457 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  21. A. A. Lunin, “Operator norms of submatrices,” Math. Notes 45 (3), 248–252 (1989) [transl. from Mat. Zametki 45 (3), 94–100 (1989)].

    Article  MathSciNet  MATH  Google Scholar 

  22. A. W. Marcus, D. A. Spielman, and N. Srivastava, “Interlacing families. II: Mixed characteristic polynomials and the Kadison–Singer problem,” Ann. Math., Ser. 2, 182 (1), 327–350 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  23. S. Nitzan, A. Olevskii, and A. Ulanovskii, “Exponential frames on unbounded sets,” Proc. Am. Math. Soc. 144 (1), 109–118 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  24. M. Talagrand, “Embedding subspaces of \(L_1\) into \(l^N_1\),” Proc. Am. Math. Soc. 108 (2), 363–369 (1990).

    MathSciNet  MATH  Google Scholar 

  25. M. Talagrand, “Embedding subspaces of \(L_p\) in \(\ell _p^N\),” in Geometric Aspects of Functional Analysis: Israel Seminar (GAFA) 1992–94 (Birkhäuser, Basel, 1995), Oper. Theory, Adv. Appl. 77, pp. 311–326.

    Chapter  Google Scholar 

  26. M. Talagrand, Upper and Lower Bounds for Stochastic Processes: Modern Methods and Classical Problems (Springer, Berlin, 2014).

    Book  MATH  Google Scholar 

  27. V. N. Temlyakov, “The Marcinkiewicz-type discretization theorems for the hyperbolic cross polynomials,” Jaen J. Approx. 9 (1–2), 37–63 (2017).

    MathSciNet  MATH  Google Scholar 

  28. V. Temlyakov, Multivariate Approximation (Cambridge Univ. Press, Cambridge, 2018), Cambridge Monogr. Appl. Comput. Math. 32.

    Book  MATH  Google Scholar 

  29. V. N. Temlyakov, “The Marcinkiewicz-type discretization theorems,” Constr. Approx. 48 (2), 337–369 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  30. V. N. Temlyakov, “Universal discretization,” J. Complexity 47, 97–109 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  31. V. N. Temlyakov, “Sampling discretization error of integral norms for function classes,” J. Complexity 54, 101408 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  32. V. N. Temlyakov, “Sampling discretization of integral norms of the hyperbolic cross polynomials,” Proc. Steklov Inst. Math. 312, 270–281 (2021) [transl. from Tr. Mat. Inst. Steklova 312, 282–293 (2021)].

    Article  MathSciNet  MATH  Google Scholar 

  33. A. Zygmund, Trigonometric Series (Univ. Press, Cambridge, 1959), Vols. 1, 2.

    MATH  Google Scholar 

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Acknowledgments

The author would like to thank Professor V. N. Temlyakov and I. V. Limonova for stimulating discussions and for providing valuable comments on the manuscript. The author would also like to thank the anonymous referee for the comments that helped to significantly improve the paper. The author is a Young Russian Mathematics award winner and would like to thank its sponsors and jury.

Funding

The work was supported by the Government of the Russian Federation, grant no. 14.W03.31.0031.

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Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    Egor D. Kosov

  2. Moscow Center for Fundamental and Applied Mathematics, Moscow, 119991, Russia

    Egor D. Kosov

  3. National Research University Higher School of Economics, Pokrovskii bul. 11, Moscow, 109028, Russia

    Egor D. Kosov

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  1. Egor D. Kosov
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Correspondence to Egor D. Kosov.

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2022, Vol. 319, pp. 202–212 https://doi.org/10.4213/tm4271.

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Kosov, E.D. Remarks on Sampling Discretization of Integral Norms of Functions. Proc. Steklov Inst. Math. 319, 189–199 (2022). https://doi.org/10.1134/S0081543822050133

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