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Transactions of the American Mathematical Society

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Some new classes of kernels whose Fredholm determinants have order less than one


Author: Dale W. Swann
Journal: Trans. Amer. Math. Soc. 160 (1971), 427-435
MSC: Primary 45.11
DOI: https://doi.org/10.1090/S0002-9947-1971-0283513-1
MathSciNet review: 0283513
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Abstract: Let $ K(s,t)$ be a complex-valued $ {L_2}$ kernel on the square $ a \leqq s,t \leqq b$ and $ \{ {\lambda _v}\} $, perhaps empty, denote the set of finite characteristic values (f.c.v.) of $ K$, arranged according to increasing modulus. Such f.c.v. are complex numbers appearing in the integral equation $ {\phi _v}(s) = {\lambda _v}\int_a^b {K(s,t){\phi _v}(t)dt} $, where the $ {\phi _v}(s)$ are nontrivial $ {L_2}$ functions on $ [a,b]$. Further let $ {k_1} = \int_a^b {K(s,s)} $ be well defined so that the Fredholm determinant of $ K,D(\lambda )$, exists, and let $ \mu $ be the order of this entire function. It is shown that (1) if $ K(s,t)$ is a function of bounded variation in the sense of Hardy-Krause, then $ \mu \leqq 1$; (2) if in addition to the assumption (1), $ K(s,t)$ satisfies a uniform Lipschitz condition of order $ \alpha > 0$ with respect to either variable, then $ \mu < 1$ and $ {k_1} = {\Sigma _v}1/{\lambda _v}$; (3) if $ K(s,t)$ is absolutely continuous as a function of two variables and $ {\partial ^2}K/\partial s\partial t$ (which exists almost everywhere) belongs to class $ {L_p}$ for some $ p > 1$, then $ \mu < 1$ and $ {k_1} = {\Sigma _v}1/{\lambda _v}$. In (2) and (3), the condition $ {k_1} \ne 0$ implies $ K(s,t)$ possesses at least one f.c.v.


References [Enhancements On Off] (What's this?)

  • [1] I. Fredholm, Sur une classe d'équations fonctionnelles, Acta Math. 27 (1903), 365-390. MR 1554993
  • [2] T. Carleman, Zur Theorie der linearen Integralgleichungen, Math. Z. 9 (1921), 196-217. MR 1544464
  • [3] D. W. Swann, Kernels with only a finite number of characteristic values, Proc. Cambridge Philos. Soc. (to appear). MR 0291743 (45:834)
  • [4] T. Lalesco, Sur l'ordre de la fonction entière $ D(\lambda )$ de Fredholm, C. R. Acad. Sci. Paris 145 (1907), 906-907.
  • [5] -, Introduction à la theorie des équations intégrales, Hermann, Paris, 1912, pp. 86-89.
  • [6] J. A. Cochran, The existence of eigenvalues for the integral equations of laser theory, Bell System Tech. J. 44 (1965), 77-88. MR 30 #1368. MR 0171137 (30:1368)
  • [7] -, The analysis of linear integral equations, McGraw-Hill, New York, 1972. MR 0447991 (56:6301)
  • [8] E. W. Hobson, The theory of functions of a real variable and the theory of Fourier's series. Vol. 1, Dover, New York, 1958. MR 19, 1166.
  • [9] J. A. Clarkson and C. R. Adams, On definitions of bounded variation for functions of two variables, Trans. Amer. Math. Soc. 35 (1933), 824-854. MR 1501718
  • [10] -, Properties of functions $ f(x,y)$ of bounded variation, Trans. Amer. Math. Soc. 36 (1934), 711-730. MR 1501762
  • [11] I. P. Natanson, Theory of functions of a real variable, GITTL, Moscow, 1950; English transl., Ungar, New York, 1955. MR 12, 598; MR 16, 804. MR 0039790 (12:598d)
  • [12] E. F. Beckenbach and R. Bellman, Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 30, Springer-Verlag, New York, 1965, p. 21. MR 33 #236. MR 0192009 (33:236)
  • [13] E. T. Copson, An introduction to the theory of functions of a complex variable, Clarendon Press, Oxford, 1935, p. 223.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0283513-1
Keywords: $ {L_2}$ kernels, characteristic values, integral equations, Fredholm determinant, Lipschitz (Hölder) conditions, functions of bounded variation in Hardy-Krause sense, absolutely continuous functions of two variables
Article copyright: © Copyright 1971 American Mathematical Society

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