Far Eastern Mathematical Journal

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On the convergence of polynomial Fredholm series

I. M. Novitskii

2009, issue 1-2, P. 131–139

Abstract

In this note, we study the infinite system of Fredholm series of polynomials in $\lambda$, formed, in the classical way, for a kernel on $\mathbb{R}^2$ of the form $\boldsymbol{H}(s,t)-\lambda\boldsymbol{S}(s,t)$, where $\lambda$ is a complex parameter. We establish a convergence of these series in the complex plane with respect to sup-norms of various spaces of continuous functions. The convergence results apply to solving a Fredholm integral equation with a kernel that is linear with respect to parameter.

Keywords: linear nuclear operator, linear integral operator, Fredholm integral equation, Fredholm series, Fredholm determinant, Fredholm minor

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References

[1] I. C. Goxberg, M. G. Krejn, Vvedenie v teoriyu linejnyx nesamosopryazhennyx operatorov v gil'bertovom prostranstve, Nauka, M., 1965, 448 s. [2] I. M. Novitskii, “Integral representations of linear operators by smooth Carleman kernels of Mercer type”, Proc. London Math. Soc. (3), 68, 1994, 161–177. [3] I. M. Novitskii, “Unitary equivalence to integral operators and an application”, Int. J. Pure Appl. Math., 50:2 (2009), 295–300. [4] I. M. Novickij, “O minorax Fredgol'ma dlya vpolne nepreryvnyx operatorov”, Dal'nevostochnyj matematicheskij sbornik, 7 (1999), 103–122, Dal'nauka, Vladivostok. [5] U. V. Lovitt, Linejnye integral'nye uravneniya, GITTL, M., 1957, 266 s. [6] M. Markus, Ch. Mink, Obzor po teorii matric i matrichnyx neravenstv, Nauka, M., 1972, 232 s. [7] I. M. Novickij, “Formuly Fredgol'ma dlya yader, linejnyx otnositel'no parametra”, Dal'nevostochnyj mat. Zhurn., 3:2 (2002), 173–194.

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