the frequencies obtained and the parity of cosine, then we get the following amplitude and frequency sum (of the same order $n$): $$\label{J_0_super} H_n(x)=\frac{1}{n}\sum_{k=1}^{n} \cos(x\cdot\cos
\tfrac{(2k-1)\pi}{4n}),$$ for which by (\[Gauss-Chebyshev-error\]) we have $$\label{J_0_super_error} J_0(x)\approx H_n(x),\qquad
|J_0(x)-H_n(x)|{\leqslant}\frac{|x|^{4n}}{2^{4n-1}(4n)!}.$$ Note that the formula (\[J\_0\_super\]) can be also obtained via (\[interpolaton\_even\]).

In [@YF Table 1 and Formula (25)] the following approximant was obtained (for the approach from [@YF] see also Section \[Section6\]): $$\label{YF_J_0_cos} J_0(x)\approx
\omega_{11}(x):=\sum_{k=1}^{11}\alpha_m \cos(\beta_m x), \quad
\alpha_k\in(0.086,0.096),\quad \beta_k\in(0,1).$$ In comparison with this approximant, the sum (\[J\_0\_super\]) gives much more precise results for $n=11$ in the segment $[0,30]$. Calculations in Maple show that $$\label{YF_J_0_cos++}
 {\rm log}_{10}\frac{|J_0(x)-\omega_{11}(x)|}{|J_0(x)-H_{11}(x)|}>
 M(x):=\frac{38}{x+1},
\qquad x\in [0;30].$$ Moreover, in the segment \[0,10\] the minorant $M(x)$ can be exchanged by the more precise $\tilde M(x)=44\log_{10}
\frac{1}{x}+50\to\infty$, $x\to 0$. Note that the absolute error of the formula (\[YF\_J\_0\_cos\]) in the segment $[0,10]$ is quite small and close to $\varepsilon=10^{-12}$.

For $30{\leqslant}x{\leqslant}40$ the absolute error of the formula (\[J\_0\_super\]) is also less than the one of (\[YF\_J\_0\_cos\]), but for $x{\geqslant}40$ the errors of both approximants can reach $10^{-1}$, thus use of them does not seem to be fair for such $x$.

From (\[2n-bis-interpol\]) and (\[J\_0\_super\]) we get the following interpolation formula for the 