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Transactions of the American Mathematical Society

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Some ramifications of a theorem of Boas and Pollard concerning the completion of a set of functions in $L^2$

Authors: K. S. Kazarian and Robert E. Zink
Journal: Trans. Amer. Math. Soc. 349 (1997), 4367-4383
MSC (1991): Primary 42B65, 42C15, 46B15, 41A30, 41A58
MathSciNet review: 1443881
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Abstract: About fifty years ago, R. P. Boas and Harry Pollard proved that an orthonormal system that is completable by the adjunction of a finite number of functions also can be completed by multiplying the elements of the given system by a fixed, bounded, nonnegative measurable function. In subsequent years, several variations and extensions of this theorem have been given by a number of other investigators, and this program is continued here. A mildly surprising corollary of one of the results is that the trigonometric and Walsh systems can be multiplicatively transformed into quasibases for $L^{1}[0,1]$.

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K. S. Kazarian
Affiliation: Departamento de Matemáticas, C-XV, Universidad Autónoma de Madrid, 28049 Madrid, Spain; Institute of Mathematics of the National Academy of Sciences, av. Marshal Bagra- mian, 24-b, 375019 Erevan, Republica Armenia

Robert E. Zink
Affiliation: Department of Mathematics, Purdue University, 1395 Mathematical Sciences Building, West Lafayette, Indiana 47907-1395,  USA

Keywords: Multiplicative completion, weighted $L^{p}$-spaces, Schauder basis, quasibasis, $M$-basis, approximate continuity
Received by editor(s): March 8, 1995
Received by editor(s) in revised form: July 21, 1995
Additional Notes: The first author was supported by DGICYT Spain, under Grant PB94-0149, and also by Grant MVR000 from the I.S.F
Article copyright: © Copyright 1997 American Mathematical Society