The Smallest Eigenvalue of Hankel Matrices
By Christian Berg, Ryszard Szwarc. 2009
56 citations recorded by the source corpus.
Let ℋN=(sn+m),0≤n,m≤N, denote the Hankel matrix of moments of a positive measure with moments of any order. We study the large N behavior of the smallest eigenvalue λN of ℋN. It is proven that λN has exponential decay to zero for any measure with compact support. For general determinate moment problems the decay to 0 of λN can be arbitrarily slow or arbitrarily fast in a sense made precise below. In the indeterminate case, where λN is known to be bounded below by a strictly positive constant, we prove that the limit of the nth smallest eigenvalue of ℋN for N→∞ tends rapidly to infinity with n. The special case of the Stieltjes–Wigert polynomials is discussed.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/b40259ceab86a599b753fd0be176f7ca577c91f7
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In Applied Mathematics
In Computational Theory and Mathematics
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Citations in the corpus
- Methodology and theory for partial least squares applied to functional data (2012, background): “In such cases the smallest eigenvalue of H can converge to zero arbitrarily fast as p diverges [Berg and Szwarc (2011), Theorem 2.5], although more is known about the case where m is a continuous than that of a discrete measure, and it is particularly challenging to develop general theory…”