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 Chebyshevskii Sb., 2018, Volume 19, Issue 3, Pages 298–310 (Mi cheb696)

On complete rational trigonometric sums and integrals

V. N. Chubarikov

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: Asymptotical formulae as $m\to\infty$ for the number of solutions of the congruence system of a form
$$g_s(x_1)+…+g_s(x_k)\equiv g_s(x_1)+…+g_s(x_k)\pmod{p^m}, 1\leq s\leq n,$$
are found, where unknowns $x_1,…,x_k,y_1,…,y_k$ can take on values from the complete system of residues modulo $p^m,$ but degrees of polynomials $g_1(x),…,g_n(x)$ do not exceed $n.$ Such polynomials $g_1(x),…,g_n(x),$ for which these asymptotics hold as $2k>0,5n(n+1)+1,$ but as $2k\leq 0,5n(n+1)+1$ the given asymptotics have no place, were shew.
Besides, for polynomials $g_1(x),…,g_n(x)$ with real coefficients, moreover degrees of polynomials do not exceed $n,$ the asymptotic of a mean value of trigonometrical integrals of the form
$$\int\limits_0^1e^{2\pi if(x)}, f(x)=\alpha_1g_1(x)+…+\alpha_ng_n(x),$$
where the averaging is lead on all real parameters $\alpha_1,…,\alpha_n,$ is found. This asymptotic holds for the power of the averaging $2k>0,5n(n+1)+1,$ but as $2k\leq 0,5n(n+1)+1$ it has no place.

Keywords: complete rational trigonometric sums, trigonometric integrals.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00071_à

DOI: https://doi.org/10.22405/2226-8383-2018-19-3-298-310

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UDC: 511.3
Accepted:15.10.2018

Citation: V. N. Chubarikov, “On complete rational trigonometric sums and integrals”, Chebyshevskii Sb., 19:3 (2018), 298–310

Citation in format AMSBIB
\Bibitem{Chu18} \by V.~N.~Chubarikov \paper On complete rational trigonometric sums and integrals \jour Chebyshevskii Sb. \yr 2018 \vol 19 \issue 3 \pages 298--310 \mathnet{http://mi.mathnet.ru/cheb696} \crossref{https://doi.org/10.22405/2226-8383-2018-19-3-298-310} \elib{https://elibrary.ru/item.asp?id=39454405}