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Systems of functions which are complete in measure

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Abstract

In this note it is proved that if a complete orthonormal system {ϕ n} in L2[0, 1] contains a subsystem {ϕ nk} of a lacunary order p>2, then for some bounded measurable function h(x) the system {h(x)ϕ n(x)}n≠nk is complete in L2[0, 1].

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Literature cited

  1. R. P. Boas and H. Pollard, “The multiplicative completion of sets of functions,” Bull. Amer. Math. Soc.,54, 518–522 (1948).

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  2. A. A. Talalyan, “Representation of measurable functions by series,” Usp. Matem. Nauk,15, No. 5, 77–141 (1960).

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  3. N. K. Bari, Trigonometrical Series [in Russian], Fizmatgiz, Moscow (1961).

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Authors and Affiliations

  1. M. V. Lomonosov Moscow State University, USSR

    N. B. Pogosyan

Authors
  1. N. B. Pogosyan
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Additional information

Translated from Matematicheskie Zametki, Vol. 18, No. 6, pp. 815–824, December, 1975.

In conclusion the author thanks E. M. Nikishin for useful discussions regarding this note.

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Pogosyan, N.B. Systems of functions which are complete in measure. Mathematical Notes of the Academy of Sciences of the USSR 18, 1075–1080 (1975). https://doi.org/10.1007/BF01099984

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  • DOI: https://doi.org/10.1007/BF01099984

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