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On wavelet polynomials and Weyl multipliers

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Abstract

For the wavelet type orthonormal systems ϕn, we establish a new bound

$${\left\| {\mathop {\max}\limits_{1 \le m \le n} \left| {\sum\limits_{j \in {G_m}} {\langle f,{\phi _j}\rangle} {\phi _j}} \right|} \right\|_p} \mathbin{\lower.3ex\hbox{$\buildrel<\over{\smash{\scriptstyle\sim}\vphantom{_x}}$}} \sqrt {\log (n + 1)} \cdot ||f||{_p},\,\,\,\,\,\,1<p<\infty,$$

where Gm ⊂ ℕ are arbitrary sets of indexes. Using this estimate, we prove that log n is an almost everywhere convergence Weyl multiplier for any orthonormal system of non-overlapping wavelet polynomials. It will also be remarked that log n is the optimal sequence in this context.

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References

  1. S.-Y. A. Chang, J. M. Wilson and T. H. Wolff, Some weighted norm inequalities concerning the Schrödinger operators, Comment. Math. Helv. 60 (1985), 62–65.

    Article  MATH  Google Scholar 

  2. G. G. Gevorkyan, On Weyl factors for the unconditional convergence of series in the Franklin system, Mat. Zametki 41 (1987), 62–65.

    MathSciNet  Google Scholar 

  3. L. Grafakos, P. Honzík and A. Seeger, On maximal functions for Mikhlin–Hórmander multipliers, Adv. Math. 204 (2006), 62–65.

    Article  MATH  Google Scholar 

  4. G. Gripenberg, Wavelet bases in Lp(R), Studia Math. 106 (1993), 175–187.

    Article  MathSciNet  MATH  Google Scholar 

  5. E. Hernández and G. Weiss, A First Course on Wavelets, CRC Press, Boca Raton, FL, 1996.

    Book  MATH  Google Scholar 

  6. S. Kačmaž and G. Šteingauz, Teoriya ortogonalnykh ryadov, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1958.

    Google Scholar 

  7. G. A. Karagulyan, On systems of non-overlapping Haar polynomials, Ark. Math. 58 (2020), 62–65.

    Article  MathSciNet  MATH  Google Scholar 

  8. G. A. Karagulyan, On Weyl multipliers of the rearranged trigonometric system, Sb. Math. 211 (2020), 62–65.

    Article  MathSciNet  MATH  Google Scholar 

  9. G. A. Karagulyan, A sharp estimate for the majorant norm of a rearranged trigonometric system, Russian Math. Surveys 75 (2020), 62–65.

    Article  MathSciNet  MATH  Google Scholar 

  10. G. A. Karagulyan and M. T. Lacey, On logarithmic bounds of maximal sparse operators, Math. Z. 294 (2020), 62–65.

    Article  MathSciNet  MATH  Google Scholar 

  11. B. S. Kashin and A. A. Saakyan, Orthogonal Series, American Mathematical Society, Providence, RI, 1989.

    MATH  Google Scholar 

  12. D. E. Menshov, Sur les series de fonctions orthogonales I, Fund. Math. 4 (1923), 62–65.

    Google Scholar 

  13. Y. Meyer Ondelettes et opérateurs. II, in Actualités Mathématiques, Hermann, Paris, 1990, pp. 217–384.

    Google Scholar 

  14. P. F. X. Müller, Isomorphisms Between H1 Spaces, Birkhäuser, Basel, 2005.

    MATH  Google Scholar 

  15. E. M. Nikišin and P.L. Ul’janov, On absolute and unconditional convergence, Uspehi Mat. Nauk 22 (1967), 240–242.

    MathSciNet  Google Scholar 

  16. A. M. Olevskiı, Divergent Fourier series, Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963), 62–65.

    MathSciNet  Google Scholar 

  17. S. N. Poleščuk, On the unconditional convergence of orthogonal series, Anal. Math. 7 (1981), 62–65.

    MathSciNet  Google Scholar 

  18. H. Rademacher, Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen, Math. Ann. 87 (1922), 62–65.

    Article  MATH  Google Scholar 

  19. E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.

    MATH  Google Scholar 

  20. P. L. Ul’janov, Weyl multipliers for the unconditional convergence of orthogonal series, Dokl. Akad. Nauk SSSR 235 (1977), 62–65.

    MathSciNet  Google Scholar 

  21. P. Wojtaszczyk, Wavelets as unconditional bases in Lp(R), J. Fourier Anal. Appl. 5 (1999), 73–85.

    Article  MathSciNet  MATH  Google Scholar 

  22. B. Wolnik, The wavelet type systems, in Approximation and Probability, Polish Academy of Sciences, Institute of Mathematics, Warsaw, 2006, pp. 397–406.

    Book  MATH  Google Scholar 

  23. A. Zygmund, Trigonometric Series. Vol. 2, Cambridge University Press, New York, 1959.

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Mathematics, Polish Academy of Sciences, ul. Abrahama 18, 80-825, Sopot, Poland

    Anna Kamont

  2. Faculty of Mathematics and Mechanics, Yerevan State University, Alex Manoogian 1, 0025, Yerevan, Armenia

    Grigori A. Karagulyan

Authors
  1. Anna Kamont
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  2. Grigori A. Karagulyan
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Corresponding author

Correspondence to Grigori A. Karagulyan.

Additional information

The second author was supported by the Science Committee of RA, in the framework of the research project 21AG-1A045.

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Kamont, A., Karagulyan, G.A. On wavelet polynomials and Weyl multipliers. JAMA 150, 529–545 (2023). https://doi.org/10.1007/s11854-023-0281-4

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  • DOI: https://doi.org/10.1007/s11854-023-0281-4

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