Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Restricting a Schauder basis to a set of positive measure


Author: James Shirey
Journal: Trans. Amer. Math. Soc. 184 (1973), 61-71
MSC: Primary 42A64
DOI: https://doi.org/10.1090/S0002-9947-1973-0330914-0
MathSciNet review: 0330914
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \{ {f_n}\} $ be an orthonormal system of functions on [0, 1] containing a subsystem $ \{ {f_{{n_k}}}\} $ for which (a) $ {f_{{n_k}}} \to 0$ weakly in $ {L_2}$, and (b) given $ E \subset [0,1]$, $ \vert E\vert > 0$, $ {\operatorname{Lim}}\;{\operatorname{Inf}}{\smallint _E}\vert{f_{{n_k}}}(x)\vert dx > 0$. There then exists a subsystem $ \{ {g_n}\} $ of $ \{ {f_n}\} $ such that for any set E as above, the linear span of $ \{ {g_n}\} $ in $ {L_1}(E)$ is not dense.

For every set E as above, there is an element of $ {L_p}(E)$, $ 1 < p < \infty $, whose Walsh series expansion converges conditionally and an element of $ {L_1}(E)$ whose Haar series expansion converges conditionally.


References [Enhancements On Off] (What's this?)

  • [D] M. M. Day, Normed linear spaces, 2nd rev. ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Heft 21, Academic Press, New York; Springer-Verlag, Berlin, 1962. MR 26 #2847.
  • [G] V. F. Gapoškin, Lacunary series and independent functions, Uspehi Mat. Nauk 21 (1966), no. 6 (132), 3-82= Russian Math. Surveys 21 (1966), no. 6, 1-82. MR 34 #6374. MR 0206556 (34:6374)
  • [K] S. Kaczmarz and H. Steinhaus, Theorie der Orthogonalreihen, reprint, Chelsea, New York, 1951. MR 20 #1148.
  • [KP] M. I. Kadec and A. Pełczyński, Bases, lacunary sequences, and complemented subspaces in the spaces $ {L_p}$, Studia Math. 21 (1961/62), 161-176. MR 27 #2851. MR 0152879 (27:2851)
  • [L] G. G. Lorentz, Bernstein polynomials, Mathematical Expositions, no. 8, Univ. of Toronto Press, Toronto, 1953. MR 15, 217. MR 0057370 (15:217a)
  • [M] Jurg T. Marti, Introduction to the theory of bases, Springer-Verlag, New York, 1959. MR 0438075 (55:10994)
  • [O] W. Orlicz, Über unbedingte Convergenz in Funktionräumen. I, Studia Math. 4 (1933), 33-37.
  • [P] R. E. A. C. Paley, A remarkable series of orthogonal functions, Proc. London Math. Soc. 34 (1931), 241-264.
  • [PZ] J. J. Price and R. E. Zink, On sets of completeness for families of Haar functions, Trans. Amer. Math. Soc. 119 (1965), 262-269. MR 32 #1499. MR 0184023 (32:1499)
  • [SZ] J. Shirey and R. E. Zink, On unconditional bases in certain Banach function spaces, Studia Math. 36 (1970), 169-175. MR 43 #899. MR 0275142 (43:899)
  • [Z] A. C. Zaanen, An introduction to the theory of integration, rev. ed., North-Holland, Amsterdam; Interscience, New York, 1967. MR 36 #5286. MR 0222234 (36:5286)
  • [Zy] A. Zygmund, Trigonometrical series, 2nd ed., reprinted with corrections and some additions, Cambridge Univ. Press, New York, 1968. MR 38 #4882. MR 0076084 (17:844d)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 42A64

Retrieve articles in all journals with MSC: 42A64


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0330914-0
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society