Geodesic
A Kernel Smoothing Method for General Integral Equations
Dalʹnevostočnyj matematičeskij žurnal, Tome 12 (2012) no. 2, pp. 255-261.

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In this paper, we reduce the general linear integral equation of the third kind in $L^2(Y,\mu)$, with largely arbitrary kernel and coefficient, to an equivalent integral equation either of the second kind or of the first kind in $L^2(\mathbb{R})$, with the kernel being the linear pencil of bounded infinitely differentiable bi-Carleman kernels expandable in absolutely and uniformly convergent bilinear series. The reduction is done by using unitary equivalence transformations. 
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I. M. Novitskii. A Kernel Smoothing Method for General Integral Equations. Dalʹnevostočnyj matematičeskij žurnal, Tome 12 (2012) no. 2, pp. 255-261. http://geodesic.mathdoc.fr/item/DVMG_2012_12_2_a10/

[1] N. I. Akhiezer, “Integral operators with Carleman kernels”, Uspekhi Mat. Nauk., 2:5 (1947), 93–132 (in Russian) | MR | Zbl

[2] N. I. Akhiezer, I. M. Glazman, Theory of linear operators in Hilbert space, Nauka, Moscow, 1966 (in Russian) | Zbl

[3] P. Auscher, G. Weiss, M. V. Wickerhauser, “Local sine and cosine bases of Coifman and Meyer and the construction of smooth wavelets”, Wavelets: a tutorial in theory and applications, ed. C. Chui, Academic Press, Boston, 1992, 237–256 | MR

[4] T. Carleman, Sur les équations intégrales singulières à noyau réel et symétrique, A.-B. Lundequistska Bokhandeln, Uppsala, 1923

[5] C. G. Costley, “On singular normal linear equations”, Can. Math. Bull., 13 (1970), 199–203 | DOI | MR | Zbl

[6] P. Halmos, V. Sunder, Bounded integral operators on $L^2$ spaces, Springer, Berlin, 1978 | MR

[7] E. Hernández, G. Weiss, A first course on wavelets, CRC Press, New York, 1996 | MR | Zbl

[8] T. T. Kadota, “Term-by-term differentiability of Mercer's expansion”, Proc. Amer. Math. Soc., 18 (1967), 69–72 | MR | Zbl

[9] V. B. Korotkov, Integral operators, Nauka, Novosibirsk, 1983 (in Russian) | MR

[10] V. B. Korotkov, “Systems of integral equations”, Sibirsk. Mat. Zh., 27:3 (1986), 121–133 (in Russian) | MR | Zbl

[11] V. B. Korotkov, “On the reduction of families of operators to integral form”, Sibirsk. Mat. Zh., 28:3 (1987), 149–151 (in Russian) | MR | Zbl

[12] V. B. Korotkov, Some questions in the theory of integral operators, Institute of Mathematics of the Siberian Branch of the Academy of Sciences of USSR, Novosibirsk, 1988 (in Russian) | Zbl

[13] J. Mercer, “Functions of positive and negative type, and their connection with the theory of integral equations”, Philos. Trans. Roy. Soc. London Ser. A, 209 (1909), 415–446 | DOI | Zbl

[14] I. M. Novitskii, “Integral representations of linear operators by smooth Carleman kernels of Mercer type”, Proc. Lond. Math. Soc. (3), 68:1 (1994), 161–177 | DOI | MR

[15] I. M. Novitskii, “Fredholm minors for completely continuous operators”, Dal'nevost. Mat. Sb., 7 (1999), 103–122 (in Russian)

[16] I. M. Novitskii, “Fredholm formulae for kernels which are linear with respect to parameter”, Dal'nevost. Mat. Zh., 3:2 (2002), 173–194 (in Russian) | MR

[17] I. M. Novitskii, “Integral representations of unbounded operators by infinitely smooth kernels”, Cent. Eur. J. Math., 3:4 (2005), 654–665 | DOI | MR | Zbl

[18] I. M. Novitskii, “Integral representations of unbounded operators by infinitely smooth bi-Carleman kernels”, Int. J. Pure Appl. Math., 54:3 (2009), 359–374 | MR | Zbl

[19] I. M. Novitskii, “On the convergence of polynomial Fredholm series”, Dal'nevost. Mat. Zh., 9:1-2 (2009), 131–139 | MR

[20] I. M. Novitskii, “Unitary equivalence to integral operators and an application”, Int. J. Pure Appl. Math., 50:2 (2009), 295–300 | MR | Zbl

[21] I. M. Novitskii, “Integral operators with infinitely smooth bi-Carleman kernels of Mercer type”, Int. Electron. J. Pure Appl. Math., 2:1 (2010), 43–73 | MR

[22] I. M. Novitskii, “Kernels of integral equations can be boundedly infinitely differentiable on $\mathbb{R}^2$”, Proc. 2011 World Congress on Engineering and Technology (CET 2011, Shanghai, China, Oct. 28–Nov. 2, 2011), v. 2, {IEEE} Press, Beijing, 2011, 789–792

[23] W. J. Trjitzinsky, “Singular integral equations with complex valued kernels”, Ann. Mat. Pura Appl., 4:25 (1946), 197–254 | DOI | MR

[24] J. von Neumann, Charakterisierung des Spektrums eines Integraloperators, Actual. scient. et industr., 229, Hermann, Paris, 1935

[25] J. W. Williams, “Linear integral equations with singular normal kernels of class I”, J. Math. Anal. Appl., 68:2 (1979), 567–579 | DOI | MR | Zbl