Eigenvalue Distribution of Positive Definite Kernels on Unbounded Domains

Abstract

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.

Eigenvalue distribution of Mercer‐like kernels

“…The class of Mercer-like kernels has been shown to specify necessary and sufficient conditions under which Mercer's theorem extends to kernels on unbounded intervals [2,3]. Using R as a model of such a domain, and bearing in mind that the results generalize in the obvious way to other types of unbounded intervals, the following summarizes the crucial properties of Mercer-like kernels [6]. …”

“…In particular, our main results will be stated for Mercer-like kernels in the following classes S n (I) (continuity or differentiability at possible boundary points is to be understood in the usual sense of one-sided limits). A number of properties of these classes have been established in [6] generalizing those essentially proved by Kadota [11] for differentiable positive definite kernels on compact domains. We mention, in particular, that in the compact domain case all the symmetric derivatives k m (x, y) ≡ ∂ 2m ∂y m ∂x m k(x, y) are automatically the kernels of trace class positive operators.…”

Operadores integrais positivos e espaços de Hilbert de reprodução

“…Os artigos de Reade [49,50] tratam do caso em que o conjunto Xé um intervalo compacto, enquanto que o trabalho de Kühn [17,38,39] trata do caso em que Xé um espaço métrico compacto ou uma variedade diferenciável compacta com medida finita. 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.…”

“…Descrevemos condições para obter resultados sobre o decaimento de autovalores do operador K, quando K satisfaz uma propriedade de suavidade do tipo Lipschitz. Enfatizamos que a metodologia usada aqui foi baseada nos trabalhos de Buescu [7,8], em [24] e em referências lá citadas. Em particular, os resultados obtidos neste capítulo produziram os artigos [25,26].…”

“…As referências ( [5,6,7,8,21,22,23,27,28,37,38,46,47,60]) contêm resultados relacionados ao assunto aqui tratado onde o Teorema de Mercer aparece como ferramenta fundamental.…”