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
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