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

DOI:
https://doi.org/10.1090/S0002-9947-97-02034-5

MathSciNet review:
1443881

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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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Additional Information

**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

Email:
kazaros.kazarian@uam.es

**Robert E. Zink**

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

Email:
zink@math.purdue.edu

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