1,123 research outputs found
Spectral properties of kernel matrices in the flat limit
Kernel matrices are of central importance to many applied fields. In this
manuscript, we focus on spectral properties of kernel matrices in the so-called
"flat limit", which occurs when points are close together relative to the scale
of the kernel. We establish asymptotic expressions for the determinants of the
kernel matrices, which we then leverage to obtain asymptotic expressions for
the main terms of the eigenvalues. Analyticity of the eigenprojectors yields
expressions for limiting eigenvectors, which are strongly tied to discrete
orthogonal polynomials. Both smooth and finitely smooth kernels are covered,
with stronger results available in the finite smoothness case.Comment: 40 pages, 8 page
On the constrained mock-Chebyshev least-squares
The algebraic polynomial interpolation on uniformly distributed nodes is
affected by the Runge phenomenon, also when the function to be interpolated is
analytic. Among all techniques that have been proposed to defeat this
phenomenon, there is the mock-Chebyshev interpolation which is an interpolation
made on a subset of the given nodes whose elements mimic as well as possible
the Chebyshev-Lobatto points. In this work we use the simultaneous
approximation theory to combine the previous technique with a polynomial
regression in order to increase the accuracy of the approximation of a given
analytic function. We give indications on how to select the degree of the
simultaneous regression in order to obtain polynomial approximant good in the
uniform norm and provide a sufficient condition to improve, in that norm, the
accuracy of the mock-Chebyshev interpolation with a simultaneous regression.
Numerical results are provided.Comment: 17 pages, 9 figure
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