Abstract
If a positive definite kernelK(x, y) has thepth order partial derivative (∂p/∂y p)K(x,y) continuous on the square [0,1]2, we show that the eigenvalues of the integral operator generated byK(x, y) are asymptoticallyo(1/n p+1). We also obtain the anticipated asymptotic estimate when (∂p/∂y p)K(x,y) satisfies further a Lipschitz condition iny of order 0<α≤1. These results, which extend some classical estimates of I. Fredholm and H. Weyl under the additional positive definiteness assumption, are based on two interesting inequalities of K. Fan.
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References
Bhatia, R., Matrix Analysis. Springer-Verlag, New York-Berlin-Heidelberg, 1996.
Cochran, J.A., Lukas, M.A., Differentiable positive definite kernels and Lipschitz continuity. Math. Proc. Camb. Phil. Soc.104 (1988), 361–369.
Coddington, E.A., Levinson, N., Theory of Ordinary Differential Equations. McGraw-Hill, New York, 1955.
Dunford, N., Schwartz, J.T.: Linear Operators, Part II: Spectral Theory. Interscience Publishers, Inc., New York 1963.
Fan, K., Maximum properties and inequalities for the eigenvalues of completely continuous operators. Proc. Nat. Acad. Sci. USA37 (1951), 760–766.
Fan, K., Inequalities for eigenvalues of Hermitian matrices. Nat. Bur. Standards Appl. Math. Ser.39 (1954), 131–139.
Fredholm, I., Sur une classe d'équations fonctionnelles. Acta Math.27 (1903), 365–390.
Gohberg, I.C., Krein, M.G., Introduction to the Theory of Linear Nonselfadjoint Operators. Transl. Math. Monographs, Vol. 18, Amer. Math. Soc., Providence, RI, 1969.
Ha, C.W., Eigenvalues of differentiable positive definite kernels. SIAM J. Math. Anal.17 (1986), 415–419.
Reade, J.B., Eigenvalues of positive definite kernels. SIAM J. Math. Anal.14 (1983), 152–157.
Weyl, H., Das Asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen. Math. Ann.71 (1912), 441–479.