Asymptotic Formula for Minimal Eigenvalues of Hilbert-type Matrices

Abstract

Let us denote by Hn = Hn(β, θ), β > 0, θ> 0, the classical Hilbert matrices:

$$ {H_n} = \left\| {{h_{j.k}}} \right\|;{h_{j,k}}(\beta ,\theta ): = \frac{{{\beta ^{j + k}}}}{{j + k + \theta }};j,k \in \{ o,1,...n\} $$

(1)

which are the Gram’s matrices for the system of powers {1, x,…, xⁿ} on the interval [0, β] with weight ω(x) = x^θ-1/ß (cf. [1], § 10.1) and play important role in various problems of approximation and extrapolation.

The work was supported by the grants of European Community (INTAS-881-94) and Russian Foundation of Basic Research (RFBR-99-01-00868).