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The Fréchet Variation, Sector Limits, and Left Decompositions

Published online by Cambridge University Press:  20 November 2018

Marston Morse  and
William Transue
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1. Introduction. The Fréchet variation of a function g defined over a 2-interval I2 was introduced by Fréchet to enable him to generalize Riesz's theorem on the representation of functionals linear over the space C [7]. Recently the authors have found this variation fundamental in the study of functionals bilinear over the Cartesian product A ⨯ B of two normed linear spaces with certain characteristic properties, and in the further use of this theory in spectral and variational analysis. The recent discovery by the authors of several new properties of the Fréchet variation has made it possible to to give new and natural tests for the convergence of multiple Fourier series generalizing the classical Jordan, de la Vallée Poussin, Dini, Young and Lebesgue tests under considerably less restrictive hypotheses than those now accepted.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 2 , 1950 , pp. 344 - 374
Copyright
Copyright © Canadian Mathematical Society 1950

References

[1] Banach, S., Théorie des opérations linéaires (Warsaw, 1932).Google Scholar
[2] Bochner, S., Summation of multiple Fourier series by spherical means, Trans. Amer. Math. Soc., vol. 40 (1936), 175–207.Google Scholar
[3] Bochner, S. and Chandrasekharan, K., On the localization property for multiple Fourier series, Ann. of Math., vol. 49 (1948), 966–978.Google Scholar
[4] Cesari, L., Sulle funzioni di due variabili a variazione limitata secondo Tonelli e sulla convergenza delle relative série doppie di Fourier, Rendiconti del Seminario délia R. Université di Roma, 1937.Google Scholar
[5] Fréchet, M., Sur les fonctionnelles bilinéaires, Trans. Amer. Math. Soc, vol. 16 (1915), 215–234.Google Scholar
[6] Gergen, J. J., Convergence criteria for double Fourier series, Trans. Amer. Math. Soc, vol. 35 (1933), 29–63.Google Scholar
[7] Morse, M. and Transue, W., Functionals of bounded Fréchet variation, Can. J. Math., vol.1 (1949), 153–165.Google Scholar
[8] Morse, M. Functionals F bilinear over the product A ⨯ B of two pseudo -normed vector spaces, I. The representation of F, Ann. of Math., vol. 50 (1949).Google Scholar
[9] Morse, M. Integral representations of bilinear functionals, Proc Nat. Acad. Sci., vol. 35 (1949), 136–143.Google Scholar
[10] Morse, M. Functionals F bilinear over the product A ⨯ B of two pseudo-normed vector spaces, II. Admissible spaces A, Ann. of Math., vol. 51 (1950).Google Scholar
[11] Morse, M. A characterization of the bilinear sums associated with the classical second variation, Annali di Matematica Pura ed Applicata Ser. 4, vol. 28 (1949).Google Scholar
[12] Morse, M. Norms of distribution functions associated with bilinear functionals (to be published ).Google Scholar
[13] Morse, M. The Fréchet variation and a generalization of the Jordan test for multiple Fourier series, Rivista Matematica, vol. 1 (1950).Google Scholar
[14] Morse, M. Calculus for Fréchet variations (to be published).Google Scholar
[15] Morse, M. The Fréchet variation and the convergence of multiple Fourier series, Proc. Nat. Acad. Sci., vol. 35 (1949).Google Scholar