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Trigonometric sums of grids of algebraic lattices with infinitely differentiable weights
E. M. Rarovaa, N. N. Dobrovol'skiiab, I. Yu. Rebrovaa, N. M. Dobrovol'skiia a Tula State Lev Tolstoy Pedagogical University (Tula)
b Tula State University (Tula)
Abstract:
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 $$ |S_{M(t),\vec\rho_\infty}(\vec{m})|\le B(r,\infty)\left(\frac{1-\delta(\vec{k}_\Lambda(\vec{m}))}{(q(\vec{n}_\Lambda(\vec{m})))^{r+1}}+O\left(\frac{q(\vec{n}_\Lambda(\vec{m}))^{r+1}\ln^{s-1}\det \Lambda(t)}{ (\det\Lambda(t))^{r+1}}\right)\right). $$
Keywords:
algebraic lattices, algebraic net, trigonometric sums of algebraic net with weights, weight functions.
Received: 09.05.2021 Accepted: 20.09.2021
Citation:
E. M. Rarova, N. N. Dobrovol'skii, I. Yu. Rebrova, N. M. Dobrovol'skii, “Trigonometric sums of grids of algebraic lattices with infinitely differentiable weights”, Chebyshevskii Sb., 22:3 (2021), 166–178
Linking options:
https://www.mathnet.ru/eng/cheb1069 https://www.mathnet.ru/eng/cheb/v22/i3/p166
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