Abstract:Readefll] has shown that positive definite kernels K(x, t) which satisfy a Lipschitz condition of order a on a bounded region have eigenvalues which are asymptotically O(l/n 1+x ). In this paper we extend this result to positive definite kernels whose symmetric derivative , and estimates based upon finite rank approximations to the kernels in question. In these latter estimates we employ the familiar piecewise linear 'hat' basis functions of approximation theory.
We study eigenvalues of positive definite kernels of L 2 integral operators on arbitrary intervals. Assuming integrability and uniform continuity of the kernel on the diagonal, we show that the eigenvalue distribution is totally determined by the smoothness of the kernel together with its decay rate at infinity along the diagonal. Moreover, the rate of decay of eigenvalues depends on both these quantities in a symmetrical way. Our main result treats all possible orders of differentiability and all possible rates of decay of the kernel; the known optimal results for eigenvalue distribution of positive definite kernels in compact intervals are particular cases.
We study eigenvalues of positive definite kernels of L 2 integral operators on arbitrary intervals. Assuming integrability and uniform continuity of the kernel on the diagonal, we show that the eigenvalue distribution is totally determined by the smoothness of the kernel together with its decay rate at infinity along the diagonal. Moreover, the rate of decay of eigenvalues depends on both these quantities in a symmetrical way. Our main result treats all possible orders of differentiability and all possible rates of decay of the kernel; the known optimal results for eigenvalue distribution of positive definite kernels in compact intervals are particular cases.
“…In fact the optimal estimates are slightly sharper: λ n = o(1/n p+1 ) for odd p and ∞ 1 n p λ n < +∞ for even p; see Ha [10] and Reade [19]. Cochran and Lukas [8] and Chang and Ha [7] derive the corresponding results for the decay rate of eigenvalues when a suitable higher-order derivative is Lip α .…”
We study eigenvalues of positive definite kernels of L 2 integral operators on unbounded real intervals. Under the assumptions of integrability and uniform continuity of the kernel on the diagonal the operator is compact and trace class. We establish sharp results which determine the eigenvalue distribution as a function of the smoothness of the kernel and its decay rate at infinity along the diagonal. The main result deals at once with all possible orders of differentiability and all possible rates of decay of the kernel. The known optimal results for eigenvalue distribution of positive definite kernels in compact intervals are particular cases. These results depend critically on a 2-parameter differential family of inequalities for the kernel which is a consequence of positivity and is a differential generalization of diagonal dominance.
“…Por outro lado, o estudo do assunto, com a retirada da compacidade de X,é feito por Buescu [6,7,8], que trata do contexto onde Xé um intervalo. Os trabalhos [15,16,18,20,24,28,35] tratam deste assunto em contextos semelhantes aos citados.…”
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