has a unique solution with the spectrum, consisting of $n$ pairwise distinct points $\lambda_1,\ldots,\lambda_n$, if and only if the leading principal minors $\Delta_k$ of order $k$ of the infinite Hankel matrix $(s_{i+j})_{i,j=0}^\infty$ satisfy the following conditions: $$\label{Hamb_condition}
\Delta_1>0, \qquad \Delta_2>0,\qquad \ldots \qquad \Delta_{n}>0,\qquad \Delta_{n+1}=\Delta_{n+2}=\ldots=0.$$ This implies that the discrete moment problem (\[SRS\]) is regularly solvable in real numbers $\lambda_k$ (this is equivalent to the fact that the polynomial (\[G\_n\]) has $n$ pairwise distinct real roots) and $\mu_k>0$ if and only if the sequence (\[ssmm\]) satisfies the first $n$ inequalities in (\[Hamb\_condition\]). Note that then the sequence $\{s_m\}_{m=0}^{2n-1}$ is called *positive*.

[**.2. Gauss and Chebyshev quadratures.**]{} Given a function $f$, analytic in a $\rho$-neighbourhood of the origin, suppose that $$F(x):=\frac{1}{x}\int_{-x}^{x}f(t)\,dt,\qquad 0<x <\rho.$$ To construct the amplitude and frequency operator $H_n(\{\mu_k\},\{\lambda_k\},f;x)$ for $F(x)$, we get the (positive) moment sequence $s_{m}=\frac{1-(-1)^{m+1}}{m+1}$, $m=\overline{0,2n-1}$, from (\[ssmm\]) and then consider the corresponding discrete moment problem (\[SRS\]). It is well-known that it is regular for any $n$, moreover, the corresponding generating polynomial (\[G\_n\]) is the Legendre polynomial $P_n$ (we write it in Rodrigues’ form) multiplied by a non-zero constant [@Krylov Ch. 7, §2]: $$G_n(x)=P_n(1) P_n(x), \qquad P_n(x):=\frac{1}{2^n
n!}\frac{d^n}{dx^n}(x^2-1)^n.$$ Therefore the frequencies $\lambda_k$ are real, pairwise distinct and 