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On a Theorem of Beurling and Livingston

Published online by Cambridge University Press:  20 November 2018

Felix E. Browder
Felix E. Browder*
Affiliation:
Institute for Advanced Study and University of Chicago
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In their paper (1), Beurling and Livingston established a generalization of the Riesz-Fischer theorem for Fourier series in Lp using a theorem on duality mappings of a Banach space B into its conjugate space B*. It is our purpose in the present paper to give another proof of this theorem by deriving it from a more general result concerning monotone mappings related to recent results on non-linear functional equations in Banach spaces obtained by the writer (2, 3, 4, 5) and G. J. Minty (6).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 17 , 1965 , pp. 367 - 372
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Beurling, A. and Livingston, A. E., A theorem on duality mappings in Banach spaces, Ark Mat., 4 (1961), 405–11.Google Scholar
2. Beurling, A. and Livingston, A. E., Non-linear elliptic boundary value problems, Bull. Amer. Math. Soc, 69 (1963), 864–76.Google Scholar
3. Beurling, A. and Livingston, A. E. Non-linear parabolic boundary value problems of arbitrary order, Bull. Amer. Math. Soc, 69 (1963), 858–861.Google Scholar
4. Beurling, A. and Livingston, A. E., Strongly non-linear parabolic boundary value problems, Amer. J. Math., 86 (1964), 339–357.Google Scholar
5. Beurling, A. and Livingston, A. E., Non-linear elliptic boundary value problems, II, Trans. Amer. Math. Soc, 115 (1965).Google Scholar
6. Minty, G. J., On a “monotonicity” method for the solution of non-linear equations in Banach spaces, Proc Nat. Acad. Sci. U.S.A., 50 (1963), 1038–1041.Google Scholar