As it can be easily checked (see (\[J\_0000\]) and (\[sinc\])), then $s_m=\frac{(2m+1)!}{(2^mm!)^2}$, $m=\overline{0,2n-1}$. Solving the corresponding problem (\[SRS\]) (in all examples considered we obtained non-negative $\lambda_k$ and real $\mu_k$) yields $$\tilde{f}(t)=\sum_{k=1}^n\mu_k h(\lambda_k t)+r_n(t),\qquad
r_n(t)=O(t^{2n}),$$ or, which is the same, $$\label{J_0_sinc} J_0(x)=\sum_{k=1}^n\mu_k \,
\textrm{sinc}(\sqrt{\lambda_k}x)+r_n(x),\qquad r_n(x)=O(x^{4n}).$$

We now compare (\[J\_0\_sinc\]) with the approximate equality $J_0(x)\approx
\sum_{k=1}^{11}\alpha_k\,\textrm{sinc}(\beta_k z)$, obtained in [@YF Table 3 and Formula (32)] (the amplitudes and frequencies were found there by a numerical method). It turns out that the amplitude and frequency sum in (\[J\_0\_sinc\]) with the same order $n=11$ gives more precise results for $x\in[0;42]$ (especially in a small neighbourhood of $x=0$). For $x{\geqslant}40$ the absolute errors of both formulas can already exceed $10^{-3}$.

The authors of [@Beylkin; @Potts] consider numerical methods for interpolation of the function $J_0$ in equidistant nodes by amplitude and frequency sums of the form $\sum_{k=1}^{n}\alpha_k
\exp(\beta_k x)$ with complex (but not purely imaginary) $\alpha_k$ and $\beta_k$ and quite fast-decreasing moduli of exponents. This enables them to get good approximants of $J_0$ on large intervals. For example, in [@Beylkin] such an approximant with $n=28$ has the error $\varepsilon{\leqslant}10^{-10}$ for $x\in[0,100\pi]$. The approximant with $n=20$, obtained in [@Potts Example 4.5] by slightly different methods, has the 