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A Closure Criterion for Orthogonal Functions

  • Ross E. Graves (a1)

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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 L 2. 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) ∊ L 2 on min (c, x) < t < max (c, x), where a ≤ c ≤ b. (a, b, and c may be infinite.)

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References

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1. Courant, R. and Hilbert, D., Methoden der Mathematischen Physik, vol. 1 (Berlin, 1931).
2. Dalzell, D. P., On the completeness of a series of normal orthogonal functions, J. London Math. Soc, vol. 20 (1945), 8793.
3. Dalzell, D. P., On the completeness of Dinïs series, J. London Math. Soc, vol. 20 (1945), 213218.
4. Kacmarz, S. and Steinhaus, H., Theorie der Orthogonalreihen (Warsaw, 1935).
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