Differentiable positive definite kernels and Lipschitz continuity

Cochran, James Alan; Lukas, Mark A.

doi:10.1017/s030500410006552x

Cited by 9 publications

(9 citation statements)

References 11 publications

(22 reference statements)

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“…[8] and [10]). Recall that, as previously observed, k m is a continuous positive definite kernel defined on the compact set [0, L] 2 .…”

Section: Eigenvalues Of Symmetric Derivativesmentioning

confidence: 99%

Eigenvalue distribution of Mercer‐like kernels

Buescu

Paixão²

2007

Mathematische Nachrichten

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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.

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“…[8] and [10]). Recall that, as previously observed, k m is a continuous positive definite kernel defined on the compact set [0, L] 2 .…”

Section: Eigenvalues Of Symmetric Derivativesmentioning

confidence: 99%

Eigenvalue distribution of Mercer‐like kernels

Buescu

Paixão²

2007

Mathematische Nachrichten

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“…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 .…”

Section: Introduction and Definitionsmentioning

confidence: 99%

Eigenvalue Distribution of Positive Definite Kernels on Unbounded Domains

Buescu¹,

Paixão

2006

Integr. equ. oper. theory

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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.

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“…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.…”

Section: Considerações Adicionaisunclassified

Operadores integrais positivos e espaços de Hilbert de reprodução

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Differentiable positive definite kernels and Lipschitz continuity

Cited by 9 publications

References 11 publications

Eigenvalue distribution of Mercer‐like kernels

Eigenvalue distribution of Mercer‐like kernels

Eigenvalue Distribution of Positive Definite Kernels on Unbounded Domains

Operadores integrais positivos e espaços de Hilbert de reprodução

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