Geodesic
On Some Functional Equations of Carleman
Canadian mathematical bulletin, Tome 15 (1972) no. 1, pp. 129-131

Voir la notice de l'article provenant de la source Cambridge University Press

The celebrated Fredholm theory of linear integral equations holds if the kernel K(x, y) or one of its iterates K (n) is bounded. Hilbert utilizing his theory of quadratic form was able to extend the theory to the kernels K(x, y) satisfying a b where k is independent of u(x).These theories were extended considerably by T. Carleman who deleted condition (b) above.Equations involving this Carleman kernel have been found useful in connection with Hermitian forms, continued fractions, Schroedinger wave equations (see [1], [2]) and more recently in scattering theory in quantum physics, etc. [3]. See also [5] for a variety of applications and extensions. 
Costley, Charles G. On Some Functional Equations of Carleman. Canadian mathematical bulletin, Tome 15 (1972) no. 1, pp. 129-131. doi: 10.4153/CMB-1972-023-7
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[1] 1. Carleman, T., La théorie des équations intégrales singulières et ses applications, Ann. Inst. H. Poincaré, 1931. Google Scholar

[2] 2. Carleman, T., Sur les équations singulières ? noyau réel et symétrique, Uppsala Univ. Arsskrift 17, 1923. Google Scholar

[3] 3. Misra, B., Speiser, D., and Targonski, G., Integral operators in the theory of scattering, Helv. Phys. Act. 36. 1963. Google Scholar

[4] 4. Riesz, M. F., Uber Système Integrierbarer Funktionen, Math. Ann., 1910. Google Scholar

[5] 5. Trjitzinsky, W. J., Singular Lebesgue-Stieltjes integral equations, Acta Math. 74 (1942), 197-310. Google Scholar

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