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Subsystems of the Walsh orthogonal system whose multiplicative completions are quasibases for $L^{p}[0,1]$, $1\leq p < +\infty $

Authors: M. G. Grigorian and Robert E. Zink
Journal: Proc. Amer. Math. Soc. 131 (2003), 1137-1149
MSC (2000): Primary 42C10
Published electronically: July 25, 2002
MathSciNet review: 1948105
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Abstract: If one discards some of the elements from the Walsh family, an ancient example of a system that serves as a Schauder basis for each of the $L^{p}$-spaces, with $1< p<+\infty $, the residual system will not be a Schauder basis for any of those spaces. Nevertheless, Price has shown that each member of a large class of such subsystems is complete on subsets of $[0,1]$ that have measure arbitrarily close to $1$. In the present work, it is shown that subsystems of this kind can be multiplicatively completed in such a way that the resulting systems are quasibases for each space $L^{p}[0,1]$, $1\le p<+\infty $, from which the earlier completeness result follows as a corollary.

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  • 1. R. P. Boas and Harry Pollard, The multiplicative completion of sets of functions, Bull. Amer. Math. Soc. 54 (1948), 518-522. MR 10:189b
  • 2. Ben-Ami Braun, On the multiplicative completion of certain basic sequences in $L^{p}$, $1<p<\infty $, Trans. Amer. Math. Soc. 176 (1973), 499-508. MR 47:2331
  • 3. Bernard R. Gelbaum, Notes on Banach spaces and bases, An. Acad. Brasil 30 (1958), 29-36. MR 20:5419
  • 4. Casper Goffman and Daniel Waterman, Basic sequences in the space of measurable functions, Proc. Amer. Math. Soc. 11 (1960), 211-213. MR 22:2886
  • 5. M. Zh. Grigoryan, Convergence almost everywhere of Walsh-Fourier series of integrable functions, Izv. Akad. Nauk. Arm. SSR Ser. Math. 18 (4) (1983), 291-304 (Russian). MR 85a:42034
  • 6. M. G. Grigorian, Convergence of Walsh-Fourier series in the $L^{1}$ metric and almost everywhere; English translation in. Soviet Math. Izv. VUZ 34 (11) (1990), 9-20.
  • 7. K. S. Kazarian and Robert E. Zink, Some ramifications of a theorem of Boas and Pollard concerning the completion of a set of functions in $L^{2}$, Trans. Amer. Math. Soc. 349 (1997), 4367-4383. MR 99d:42050
  • 8. N. Levinson, Gap and Density Theorems, Amer. Math. Soc. Colloq. Publ. 26, Amer. Math. Soc., Providence, 1940. MR 2:180d
  • 9. R. E. A. C. Paley, A remarkable set of orthogonal functions, Proc. London Math. Soc. 34 (1932), 241-279.
  • 10. J. J. Price, A density theorem for Walsh functions, Proc. Amer. Math. Soc. 18 (1967), 209-211. MR 35:656
  • 11. J. J. Price and Robert E. Zink, On sets of functions that can be multiplicatively completed, Ann. Math. 82 (1965), 139-145. MR 31:1349
  • 12. Ivan Singer, Bases in Banach Spaces II, Springer Verlag, Berlin, Heidelberg, New York, 1981. MR 82k:46024
  • 13. A. A. Talalyan, On the convergence almost everywhere of subsequences of partial sums of general orthogonal series, Izv. Akad. Nauk. Arm. SSR Izv. Fiz.-Mat. Tehn Nauki 10 (1957), 17-34. MR 19:742b
  • 14. -, The representation of measurable functions by series; English translation in Russian Math. Surveys 15 (1960), 77-136.

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Additional Information

M. G. Grigorian
Affiliation: Department of Mathematics, Erevan State University, Alex Manoogian Str., 375049 Yerevan, Armenia

Robert E. Zink
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1968

Received by editor(s): August 22, 2001
Received by editor(s) in revised form: November 2, 2001
Published electronically: July 25, 2002
Communicated by: Andreas Seeger
Article copyright: © Copyright 2002 American Mathematical Society

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