Subsystems of the Walsh orthogonal system whose multiplicative completions are quasibases for ,
Authors:
M. G. Grigorian and Robert E. Zink
Journal:
Proc. Amer. Math. Soc. 131 (2003), 11371149
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 spaces, with , 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 that have measure arbitrarily close to . 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 , , from which the earlier completeness result follows as a corollary.
 1.
R.
P. Boas Jr. and Harry
Pollard, The multiplicative completion of sets
of functions, Bull. Amer. Math. Soc. 54 (1948), 518–522.
MR
0026703 (10,189b), http://dx.doi.org/10.1090/S000299041948090292
 2.
BenAmi
Braun, On the multiplicative completion of
certain basic sequences in 𝐿^{𝑝},
1<𝑝<∞, Trans. Amer. Math.
Soc. 176 (1973),
499–508. MR 0313777
(47 #2331), http://dx.doi.org/10.1090/S00029947197303137779
 3.
Bernard
R. Gelbaum, Notes on Banach spaces and bases, An. Acad.
Brasil. Ci. 30 (1958), 29–36. MR 0098974
(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 0112028
(22 #2886), http://dx.doi.org/10.1090/S00029939196001120284
 5.
M.
Zh. Grigoryan, Convergence almost everywhere of FourierWalsh
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
(85a:42034)
 6.
M. G. Grigorian, Convergence of WalshFourier series in the metric and almost everywhere; English translation in. Soviet Math. Izv. VUZ 34 (11) (1990), 920.
 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
𝐿², Trans. Amer. Math. Soc.
349 (1997), no. 11, 4367–4383. MR 1443881
(99d:42050), http://dx.doi.org/10.1090/S0002994797020345
 8.
Norman
Levinson, Gap and Density Theorems, American Mathematical
Society Colloquium Publications, v. 26, American Mathematical Society, New
York, 1940. MR
0003208 (2,180d)
 9.
R. E. A. C. Paley, A remarkable set of orthogonal functions, Proc. London Math. Soc. 34 (1932), 241279.
 10.
J.
J. Price, A density theorem for Walsh
functions, Proc. Amer. Math. Soc. 18 (1967), 209–211. MR 0209760
(35 #656), http://dx.doi.org/10.1090/S00029939196702097607
 11.
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 0177085
(31 #1349)
 12.
Ivan
Singer, Bases in Banach spaces. II, Editura Academiei
Republicii Socialiste România, Bucharest; SpringerVerlag, BerlinNew
York, 1981. MR
610799 (82k:46024)
 13.
A.
A. Talalyan, On the convergence almost everywhere of subsequences
of partial sums of general orthogonal series, Akad. Nauk Armyan. SSR.
Izv. Fiz.Mat. Estest. Tehn. Nauki 10 (1957), no. 3,
17–34 (Russian, with Armenian summary). MR 0089940
(19,742b)
 14.
, The representation of measurable functions by series; English translation in Russian Math. Surveys 15 (1960), 77136.
 1.
 R. P. Boas and Harry Pollard, The multiplicative completion of sets of functions, Bull. Amer. Math. Soc. 54 (1948), 518522. MR 10:189b
 2.
 BenAmi Braun, On the multiplicative completion of certain basic sequences in , , Trans. Amer. Math. Soc. 176 (1973), 499508. MR 47:2331
 3.
 Bernard R. Gelbaum, Notes on Banach spaces and bases, An. Acad. Brasil 30 (1958), 2936. MR 20:5419
 4.
 Casper Goffman and Daniel Waterman, Basic sequences in the space of measurable functions, Proc. Amer. Math. Soc. 11 (1960), 211213. MR 22:2886
 5.
 M. Zh. Grigoryan, Convergence almost everywhere of WalshFourier series of integrable functions, Izv. Akad. Nauk. Arm. SSR Ser. Math. 18 (4) (1983), 291304 (Russian). MR 85a:42034
 6.
 M. G. Grigorian, Convergence of WalshFourier series in the metric and almost everywhere; English translation in. Soviet Math. Izv. VUZ 34 (11) (1990), 920.
 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 , Trans. Amer. Math. Soc. 349 (1997), 43674383. 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), 241279.
 10.
 J. J. Price, A density theorem for Walsh functions, Proc. Amer. Math. Soc. 18 (1967), 209211. MR 35:656
 11.
 J. J. Price and Robert E. Zink, On sets of functions that can be multiplicatively completed, Ann. Math. 82 (1965), 139145. 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), 1734. MR 19:742b
 14.
 , The representation of measurable functions by series; English translation in Russian Math. Surveys 15 (1960), 77136.
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Additional Information
M. G. Grigorian
Affiliation:
Department of Mathematics, Erevan State University, Alex Manoogian Str., 375049 Yerevan, Armenia
Email:
gmarting@ysu.am
Robert E. Zink
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 479071968
Email:
zink@math.purdue.edu
DOI:
http://dx.doi.org/10.1090/S0002993902066182
PII:
S 00029939(02)066182
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
