### A certain class of quadratures.

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Accurate estimates of real Pochhammer products, lower (falling) and upper (rising), are presented. Double inequalities comparing the Pochhammer products with powers are given. Several examples showing how to use the established approximations are stated.

Asymptotic expressions for remainder terms of the mid-point, trapezoid and Simpson’s rules are given. Corresponding formulas with finite sums are also given.

We perform a complete study of the truncation error of the Jacobi-Anger series. This series expands every plane wave ${\mathrm{e}}^{i\widehat{s}\xb7\overrightarrow{v}}$ 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 $\u03f5$ as soon as $L$ satisfies $L+\frac{1}{2}\simeq v+C{W}^{\frac{2}{3}}\left(K{\u03f5}^{-\delta}{v}^{\gamma}\right)\phantom{\rule{0.166667em}{0ex}}{v}^{\frac{1}{3}}$ where $W$ is the Lambert function and $C\phantom{\rule{0.166667em}{0ex}},K,\phantom{\rule{0.166667em}{0ex}}\delta ,\phantom{\rule{0.166667em}{0ex}}\gamma $ are pure positive constants. Numerical experiments show that this asymptotic is optimal. Those results...

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 $\u03f5$ as soon as $L$ satisfies $L+\frac{1}{2}\simeq v+C{W}^{\frac{2}{3}}\left(K\left(\alpha \right){\u03f5}^{-\delta}{v}^{\gamma}\right)\phantom{\rule{0.166667em}{0ex}}{v}^{\frac{1}{3}}$ where $W$ is the Lambert function, $K\left(\alpha \right)$ depends only on $\alpha =\frac{|\overrightarrow{u}|}{|\overrightarrow{v}|}$ and $C\phantom{\rule{0.166667em}{0ex}},\delta ,\phantom{\rule{0.166667em}{0ex}}\gamma $ are...

We perform a complete study of the truncation error of the Jacobi-Anger series. This series expands every plane wave ${\mathrm{e}}^{i\widehat{s}\xb7\overrightarrow{v}}$ 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 ϵ as soon as L satisfies $L+\frac{1}{2}\simeq v+C{W}^{\frac{2}{3}}\left(K{\u03f5}^{-\delta}{v}^{\gamma}\right)\phantom{\rule{0.166667em}{0ex}}{v}^{\frac{1}{3}}$ where W is the Lambert function and $C\phantom{\rule{0.166667em}{0ex}},K,\phantom{\rule{0.166667em}{0ex}}\delta ,\phantom{\rule{0.166667em}{0ex}}\gamma $ are pure positive constants. Numerical experiments show that this asymptotic is...

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 ϵ as soon as L satisfies $L+\frac{1}{2}\simeq v+C{W}^{\frac{2}{3}}\left(K\left(\alpha \right){\u03f5}^{-\delta}{v}^{\gamma}\right)\phantom{\rule{0.166667em}{0ex}}{v}^{\frac{1}{3}}$ where W is the Lambert function, $K\left(\alpha \right)$ depends only on...

We obtain some approximate identities whose accuracy depends on the bottom of the discrete spectrum of the Laplace-Beltrami operator in the automorphic setting and on the symmetries of the corresponding Maass wave forms. From the geometric point of view, the underlying Riemann surfaces are classical modular curves and Shimura curves.

In the paper we present a derivative-free estimate of the remainder of an arbitrary interpolation rule on the class of entire functions which, moreover, belong to the space ${L}_{(-\infty ,+\infty )}^{2}$. The theory is based on the use of the Paley-Wiener theorem. The essential advantage of this method is the fact that the estimate of the remainder is formed by a product of two terms. The first term depends on the rule only while the second depends on the interpolated function only. The obtained estimate of the remainder of...

The aim of the paper is to get an estimation of the error of the general interpolation rule for functions which are real valued on the interval $[-a,a]$, $a\in (0,1)$, have a holomorphic extension on the unit circle and are quadratic integrable on the boundary of it. The obtained estimate does not depend on the derivatives of the function to be interpolated. The optimal interpolation formula with mutually different nodes is constructed and an error estimate as well as the rate of convergence are obtained. The general...