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On the closure of

Published online by Cambridge University Press:  24 October 2008

W. H. J. Fuchs
W. H. J. Fuchs
Affiliation:
University CollegeSwansea
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Extract

1. A set of functions {øν (t)} (ν = 1, 2, …) is said to be closed L2 in (a, b), if

implies f (t) = 0 p.p. in (a, b). It is well known that a set of functions is closed L2, if and only if every function of L2 (a, b) can be approximated in the mean square sense as closely as desired by finite linear combinations of the øν (t).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 42 , Issue 2 , June 1946 , pp. 91 - 105
Copyright
Copyright © Cambridge Philosophical Society 1946

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References

† Szàsz, O., Math. Ann. 77 (1916), 482–96.CrossRefGoogle Scholar

‡ Hardy, G. H., Messenger of Math. 46 (1917), 175–82.Google Scholar

§ Boas, R. P., Trans. American Math. Soc. 46 (1939), 142–50.CrossRefGoogle Scholar

† See, for example, Widder, D. V., The Laplace Transform, pp. 246–7.Google Scholar

† See Levinson, N., Gap and Density Theorems 1 (New York, 1940), p. 425.Google Scholar

‡ This follows from Lemma A by the limiting process T → ∞.

† Nevanlinna, R., Eindeutige analytische Funktionen (Berlin, 1936), pp. 89–92.CrossRefGoogle Scholar