A Kernel Smoothing Method for General Integral Equations |

I. M. Novitskii |

2012, issue 2, Ñ. 255–261 |

Abstract |

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. |

linear integral equations of the first, second, and third kind, unitary operator, multiplication operator, bi-integral operator, bi-Carleman kernel, Hilbert-Schmidt kernel, bilinear series expansions of kernelsKeywords: |

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## References |

[1] N. I. Akhiezer, “Integral operators with Carleman kernels”, Uspekhi Mat. Nauk., 2:5 (1947), 93–132 (in Russian). [2] N. I. Akhiezer, I. M. Glazman, Theory of linear operators in Hilbert space, Nauka, Moscow, 1966 (in Russian). [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. [4] T. Carleman, Sur les e?quations inte?grales singulie?res a? noyau re?el et syme?trique, A.-B. Lundequistska Bokhandeln, Uppsala, 1923. [5] C. G. Costley, “On singular normal linear equations”, Can. Math. Bull., 13 (1970), 199–203. [6] P. Halmos, V. Sunder, Bounded integral operators on $L^2$ spaces, Springer, Berlin, 1978. [7] E. Herna?ndez, G. Weiss, A first course on wavelets, CRC Press, New York, 1996. [8] T. T. Kadota, “Term-by-term differentiability of Mercer's expansion”, Proc. Amer. Math. Soc., 18 (1967), 69–72. [9] V. B. Korotkov, Integral operators, Nauka, Novosibirsk, 1983 (in Russian). [10] V. B. Korotkov, “Systems of integral equations”, Sibirsk. Mat. Zh., 27:3 (1986), 121–133 (in Russian). [11] V. B. Korotkov, “On the reduction of families of operators to integral form”, Sibirsk. Mat. Zh., 28:3 (1987), 149–151 (in Russian). [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). [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. [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. [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). [17] I. M. Novitskii, “Integral representations of unbounded operators by infinitely smooth kernels”, Cent. Eur. J. Math., 3:4 (2005), 654–665. [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. [19] I. M. Novitskii, “On the convergence of polynomial Fredholm series”, Dal'nevost. Mat. Zh., 9:1-2 (2009), 131–139. [20] I. M. Novitskii, “Unitary equivalence to integral operators and an application”, Int. J. Pure Appl. Math., 50:2 (2009), 295–300. [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. [22] I. M. Novitskii, “Kernels of integral equations can be boundedly infinitely differentiable on R2”, 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. [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. |