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.
Let k(x, y) be the positive definite kernel of an integral operator on an unbounded interval of R. If k belongs to class A defined below, the corresponding operator is compact and trace class. We establish two results relating smoothness of k and its decay rate at infinity along the diagonal with the decay rate of the eigenvalues. The first result deals with the Lipschitz case; the second deals with the uniformly C 1 case. The optimal results known for compact intervals are recovered as special cases, and the relevance of these results for Fourier transforms is pointed out.
Mathematics Subject Classification (2000). Primary: 45C05; 45P05.
Let I ⊆ R be a interval and k : I 2 → C be a reproducing kernel on I . By the Moore-Aronszajn theorem, every finite matrix k(x i , x j ) is positive semidefinite. We show that, as a direct algebraic consequence, if k(x, y) is appropriately differentiable it satisfies a 2-parameter family of differential inequalities of which the classical diagonal dominance is the order 0 case. An application of these inequalities to kernels of positive integral operators yields optimal Sobolev norm bounds.
In this paper we define classes of functions which we call positive definite kernel functions and positive definite kernels. The first class may be thought of as a generalization to two dimensions of the classical positive definite functions of Bochner-Khinchin type. We study their properties in depth and show how the second class arises by considering the associated integral operators. We give necessary and sufficient conditions for the existence of a bilinear expansion of Mercer type and show the analog of Bochner's theorem in the L 2 setting, namely that a function is a positive definite kernel if and only if its Fourier transform is a positive definite kernel. A simple and elegant sufficient condition for compactness of support of positive definite kernels is given, namely that they are compactly supported along the main diagonal. Several corollaries relating compactness of support of the Fourier transform and analyticity are derived.
Cutinase in aqueous solution at pH 4.5 deactivates following a parallel pathway. At 53 degrees C, 88% of the cutinase molecules are in the unfolded conformation, which can aggregate with a reaction order of 3 if the protein concentration is high (>/=12 microM). The aggregates show a sixfold increase in size as determined by dynamic light scattering. This aggregation process is the first phase observed during a deactivation experiment; however, after significant cutinase depletion and maturation of the aggregates, a first-order step starts to dominate and a second phase independent of the protein concentration is observed. Kinetic partitioning between aggregation and first-order irreversible changes of the unfolded conformation can occur during enzyme deactivation when the equilibrium between the native and the unfolded conformation is shifted and kept toward the unfolded conformation.
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