Trigonometric sums of grids of algebraic lattices with infinitely differentiable weights

E. M. Rarova ; N. N. Dobrovol'skii ; I. Yu. Rebrova ; N. M. Dobrovol'skii

Čebyševskij sbornik, Tome 22 (2021) no. 3, pp. 166-178

Résumé

The paper continues the authors' research on the evaluation of trigonometric sums of an algebraic grid with weights. The case of an arbitrary weight function of infinite order is considered. For the parameter $\vec{m}$ of the trigonometric sum $S_{M(t),\vec\rho_\infty} (\vec m)$, three cases are highlighted. If $\vec{m}$ belongs to the algebraic lattice $\Lambda (t \cdot T(\vec a))$, then for any natural $r$ the asymptotic formula is valid $$ S_{M(t),\vec\rho_\infty}(t(m,\ldots, m))=1+O\left(\frac{\ln^{s-1}\det \Lambda(t)} { (\det\Lambda(t))^{r+1}}\right). $$ If $\vec{m}$ does not belong to the algebraic lattice $\Lambda(t\cdot T(\vec a))$, then two vectors are defined $\vec{n}_\Lambda(\vec{m})=(n_1,\ldots,n_s)$ and $\vec{k}_\Lambda(\vec{m})$ from the conditions $\vec{k}_\Lambda(\vec{m})\in\Lambda$, $\vec{m}=\vec{n}_\Lambda(\vec{M})+\vec{K}_\lambda(\vec{m})$ and the product $q(\vec{n}_\lambda(\vec{m}))=\overline{n_1}\cdot\ldots\cdot\overline{n_s}$ is minimal. Asymptotic estimation is proved $$ |S_{M(t),\vec\rho_\infty}(\vec{m})|\le B(r,\infty)\left(\frac{1-\delta(\vec{k}_\Lambda(\vec{m}))}{(q(\vec{n}_\Lambda(\vec{m})))^{r+1}}+O\left(\frac{q(\vec{n}_\Lambda(\vec{m}))^{r+1}\ln^{s-1}\det \Lambda(t)}{ (\det\Lambda(t))^{r+1}}\right)\right). $$

Détail

Keywords: algebraic lattices, algebraic net, trigonometric sums of algebraic net with weights, weight functions.