The smallest eigenvalue of Hankel matrices

Abstract

Let H_N=(s_{n+m}),n,m\le N denote the Hankel matrix of moments of a positive measure with moments of any order. We study the large N behaviour of the smallest eigenvalue lambda_N of H_N. It is proved that lambda_N has exponential decay to zero for any measure with compact support. For general determinate moment problems the decay to 0 of lambda_N can be arbitrarily slow or arbitrarily fast. In the indeterminate case, where lambda_N is known to be bounded below by a positive constant, we prove that the limit of the n'th smallest eigenvalue of H_N for N tending to infinity tends rapidly to infinity with n. The special case of the Stieltjes-Wigert polynomials is discussed.

References in corpus (1)

• Fibonacci numbers and orthogonal polynomials

Cited by in corpus (14)

• Methodology and theory for partial least squares applied to functional data
• Quantum limit to subdiffraction incoherent optical imaging
• The recoverability limit for superresolution via sparsity
• On the order of indeterminate moment problems
• Self-adjoint operators associated with Hankel moment matrices
• The smallest eigenvalue of large Hankel matrices generated by a singularly perturbed Laguerre weight
• The Smallest Eigenvalue of Large Hankel Matrices Generated by a Deformed Laguerre Weight
• A few remarks on orthogonal polynomials
• Inverse of Infinite Hankel Moment Matrices
• Conditioning bounds for traveltime tomography in layered media
• Hankel determinants of random moment sequences
• Thermodynamics of the five-vertex model with scalar-product boundary conditions
• A question by Chihara about shell polynomials and indeterminate moment problems
• The Smallest Eigenvalue of Large Hankel Matrices