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A Closure Criterion for OrthogonalFunctions

Published online by Cambridge University Press:  20 November 2018

Ross E. Graves
Ross E. Graves*
Affiliation:
The University of Minnesota
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In this paper we give a simple, necessary, and sufficient condition for a sequence of orthogonal functions to be closed in L2. In theory the question of closure is reduced to the evaluation of certain integrals and the summation of an infinite series whose terms depend only upon the index n. Our principal result is

Let p(t) be a function whose zeros and discontinuities have Jordan content zero, such that for each x ∊ (a, b), p(t) ∊ L2 on min (c, x) < t < max (c, x), where a ≤ c ≤ b. (a, b, and c may be infinite.)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 4 , 1952 , pp. 198 - 203
Copyright
Copyright © Canadian Mathematical Society 1952

References

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4. Kacmarz, S. and Steinhaus, H., Theorie der Orthogonalreihen (Warsaw, 1935).Google Scholar