and $\lambda_k$ are not given but uniquely determined by the functions $f$ and $h$. Furthermore, in different applications we regard amplitude and frequency sums as operators with fixed (universal) numbers $\mu_k$ and $\lambda_k$, being determined by the analytic nature of these operators.

Examples
========

In this section we give several examples of approximating amplitude and frequency sums for some special functions, in particular, Bessel functions (all arising discrete moment problems are regularly solvable). We will also compare our approximants with similar ones, obtained by other authors.

It is known [@Abramovitz §9.1.20-21] that the Bessel function of order zero, $J_0$, has the representation $$J_0(\pm x)=\frac{1}{\pi}\int_{-x}^{x}
\frac{\exp(it)}{\sqrt{x^2-t^2}}\,dt=
\frac{1}{\pi}\int_{-x}^{x}\frac{\cos(t)}{\sqrt{x^2-t^2}}\,dt, \qquad
x>0.$$ From this, (\[Gauss-Chebyshev\]) and (\[Gauss-Chebyshev-error\]) we get the following high-accuracy local approximation of $J_0$ at the point $x=0$ by an amplitude and frequency sum of order $n$: $$\label{J_0} J_0(x)=\frac{1}{n}\sum_{k=1}^n \cos(x\cdot\cos
\tfrac{(2k-1)\pi }{2n})+r_n(x), \qquad |r_n(x)|{\leqslant}\frac{|x|^{2n}}{2^{2n-1}(2n)!}.$$ Note that one can obtain the same formula by interpolation of the series [@Abramovitz §9.1.10] $$\label{J_0000}
J_0(z)=\sum_{m=0}^\infty\frac{(-1)^m}{(2^mm!)^2}z^{2m}$$ by amplitude and frequency sums with the basis function $h(z)=\exp(z)$. Furthermore, then $$s_{2m}=\frac{(2m)!}{(2^m m!)^2},\qquad s_{2m+1}=0,\qquad
 m=\overline{1,n-1},$$ cf. $\{s_{2m}\}$ for the Gauss-Chebyshev quadrature in Section 2.2.

If in (\[J\_0\]) we use $2n$ instead of $n$ and take into account the symmetry of 