belonging to the interval $(-1,1)$ (as roots of the Legendre polynomials, forming an orthogonal system on $[-1,1]$). The amplitudes $\mu_k$ are determined via the numbers $\lambda_k$ by the well-known formulas [@Krylov Ch. 10, §3]: $$\mu_k = \frac{2}{\left( 1-\lambda_k^2 \right)
[P'_n(\lambda_k)]^2}>0.$$ Thus we obtain the interpolation formula $$\label{GAUSS}
\frac{1}{x}\int_{-x}^{x}f(t)\,dt=\sum_{k=1}^n\mu_kf(\lambda_kx)+r_n(x),
\qquad r_n(x)=O(x^{2n}),$$ which is a Gaussian quadrature for each fixed $x$. The amplitudes and frequencies depend only on $n$ but not on $f$. It is known [@Krylov Ch. 10] that the Gaussian quadratures are of the highest algebraic degree of accuracy among all the formulas of the form (\[GAUSS\]) and exact (i.e. $r_n(x)\equiv 0$) for the polynomials of degree ${\leqslant}2n-1$.

In a similar manner one can obtain interpolation formulas for integrals with classical weights. For example, for $$F(x):=\int_{-x}^{x}\frac{f(t)}{\sqrt{x^2-t^2}}\,dt,\qquad 0< x
<\rho,$$ we have $$s_{2m}=\pi\frac{(2m)!}{(2^m m!)^2},\quad s_{2m+1}=0,\quad
m=\overline{0,n-1},\qquad G_n(x)=(-2^{1-n}\pi)^n T_n(x),$$ where $T_n(x)=\cos(n\arccos x)$ are the Chebyshev polynomials of the first kind. Calculating the amplitudes $\mu_k$ via the frequencies $\lambda_k$ leads to the following Gauss-Chebyshev quadrature [@Abramovitz §25.4.38] for real $x$, $0<x <\rho$: $$\label{Gauss-Chebyshev}
\int_{-x}^{x}\frac{f(t)}{\sqrt{x^2-t^2}}\,dt=\frac{\pi}{n}
\sum_{k=1}^n f(\lambda_kx)+r_n(x), \quad \lambda_k=\cos
\tfrac{2k-1}{2n}\pi, \quad r_n(x)=O(x^{2n}),$$ whose characteristic property is the equality of the amplitudes $\mu_k=\pi/n$. The remainder can be written more precisely: $$\label{Gauss-Chebyshev-error} r_n(x)=\frac{\pi
f^{(2n)}(\xi)}{2^{2n-1}(2n)!}x^{2n},\qquad \xi\in(-x,x).$$ One 