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

Grigorian, M.; Zink, Robert

doi:10.1090/S0002-9939-02-06618-2

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

by M. G. Grigorian and Robert E. Zink PDF

Proc. Amer. Math. Soc. 131 (2003), 1137-1149 Request permission

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.

References

Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
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 313777, DOI 10.1090/S0002-9947-1973-0313777-9
Bernard R. Gelbaum, Notes on Banach spaces and bases, An. Acad. Brasil. Ci. 30 (1958), 29–36. MR 98974
Casper Goffman and Daniel Waterman, Basic sequences in the space of measurable functions, Proc. Amer. Math. Soc. 11 (1960), 211–213. MR 112028, DOI 10.1090/S0002-9939-1960-0112028-4
M. Zh. Grigoryan, Convergence almost everywhere of Fourier-Walsh series of integrable functions, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 18 (1983), no. 4, 291–304 (Russian, with English and Armenian summaries). MR 723562
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.
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), no. 11, 4367–4383. MR 1443881, DOI 10.1090/S0002-9947-97-02034-5
Leonard Eugene Dickson, New First Course in the Theory of Equations, John Wiley & Sons, Inc., New York, 1939. MR 0000002
R. E. A. C. Paley, A remarkable set of orthogonal functions, Proc. London Math. Soc. 34 (1932), 241–279.
J. J. Price, A density theorem for Walsh functions, Proc. Amer. Math. Soc. 18 (1967), 209–211. MR 209760, DOI 10.1090/S0002-9939-1967-0209760-7
J. J. Price and Robert E. Zink, On sets of functions that can be multiplicatively completed, Ann. of Math. (2) 82 (1965), 139–145. MR 177085, DOI 10.2307/1970566
Ivan Singer, Bases in Banach spaces. II, Editura Academiei Republicii Socialiste România, Bucharest; Springer-Verlag, Berlin-New York, 1981. MR 610799
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
—, The representation of measurable functions by series; English translation in Russian Math. Surveys 15 (1960), 77–136.