We perform a complete study of the truncation error of the Gegenbauer series. This series yields an expansion of the Green kernel of the Helmholtz equation, $\frac{{\mathrm{e}}^{i|\overrightarrow{u}-\overrightarrow{v}|}}{4\pi i|\overrightarrow{u}-\overrightarrow{v}|}$, which is the core of the Fast Multipole Method for the integral equations. We consider the truncated series where the summation is performed over the indices $\ell \le L$. We prove that if $v=|\overrightarrow{v}|$ is large enough, the truncated series gives rise to an error lower than as soon as satisfies $L+\frac{1}{2}\simeq v+C{W}^{\frac{2}{3}}\left(K\left(\alpha \right){ϵ}^{-\delta }{v}^{\gamma }\right)v^{\frac{1}{3}}$ where  is the Lambert function, $K\left(\alpha \right)$ depends...

We perform a complete study of the truncation error of the Jacobi-Anger series. This series expands every plane wave ${\mathrm{e}}^{i\widehat{s}·\overrightarrow{v}}$ in terms of spherical harmonics ${\left\{Y_{\ell ,m}\left(\widehat{s}\right)\right\}}_{\left|m\right|\le \ell \le \infty }$. We consider the truncated series where the summation is performed over the $(\ell ,m)$'s satisfying $\left|m\right|\le \ell \le L$. We prove that if $v=|\overrightarrow{v}|$ is large enough, the truncated series gives rise to an error lower than as soon as satisfies $L+\frac{1}{2}\simeq v+C{W}^{\frac{2}{3}}\left(K{ϵ}^{-\delta }{v}^{\gamma }\right)v^{\frac{1}{3}}$ where is the Lambert function and $C,K,\delta ,\gamma$ are pure positive constants. Numerical experiments show that this asymptotic...

This paper deals with lower bounds on the approximability of different subproblems of the Traveling Salesman Problem (TSP) which is known not to admit any polynomial time approximation algorithm in general (unless $𝒫=\mathrm{𝒩𝒫}$). First of all, we present an improved lower bound for the Traveling Salesman Problem with Triangle Inequality, -TSP for short. Moreover our technique, an extension of the method of Engebretsen [11], also applies to the case of relaxed and sharpened triangle inequality, respectively,...