The Smallest Eigenvalue of Hankel Matrices

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.

[1]  Yang Chen,et al.  Smallest eigenvalues of Hankel matrices for exponential weights , 2004 .

[2]  D. V. Widder,et al.  Review: J. A. Shohat and J. D. Tamarkin, The problem of moments , 1945 .

[3]  Yang Chen,et al.  Small eigenvalues of large Hankel matrices , 1999, math/0009238.

[4]  A. R. Collar.  XIX.—On the Reciprocation of Certain Matrices , 1940 .

[5]  J. S. Christiansen,et al.  The moment problem associated with the Stieltjes–Wigert polynomials , 2003 .

[6]  Yang Chen,et al.  Small eigenvalues of large Hankel matrices: The indeterminate case , 1999 .

[7]  Doron S. Lubinsky,et al.  Condition numbers of Hankel matrices for exponential weights , 2006 .

[8]  Bernhard Beckermann,et al.  The condition number of real Vandermonde, Krylov and positive definite Hankel matrices , 2000, Numerische Mathematik.

[9]  Stefan Rolewicz,et al.  On a problem of moments , 1968 .

[10]  Herbert S. Wilf,et al.  Finite Sections of Some Classical Inequalities , 1970 .

[11]  G. Szegö,et al.  On some Hermitian forms associated with two given curves of the complex plane , 1936 .

[12]  M. Schlosser.  BASIC HYPERGEOMETRIC SERIES , 2007 .

[13]  T. Chihara.  CHAIN SEQUENCES AND ORTHOGONAL POLYNOMIALS , 1962 .

[14]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[15]  The index of determinacy for measures and the ²-norm of orthonormal polynomials , 1995 .

[16]  B. Simon.  The Classical Moment Problem as a Self-Adjoint Finite Difference Operator , 1998, math-ph/9906008.

[17]  C. Berg,et al.  Orthogonal polynomials and analytic functions associated to positive definite matrices , 2006 .

[18]  Fibonacci numbers and orthogonal polynomials , 2006, math/0609283.

[19]  Christian Berg,et al.  Density questions in the classical theory of moments , 1981 .

[20]  T. Stieltjes.  Recherches sur les fractions continues , 1995 .

[21]  Barry Simon,et al.  The Christoffel-Darboux Kernel , 2008, 0806.1528.