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Тезисы докладов

Аппроксимация функций и квадратурные формулы

One of the main problems of the lattice cubature formulas theory is to get best estimates for the error functional begin{equation} label{n1} l:f to (l,f)equiv int_Omega f(x)dx -h^nsum_{hHk}inOmega c_k(h)f(hHk) end{equation} At this formula $Omega$ is boundary domain in $mathbb R^n $, H-nxn matrix with det$H=1$, ${hHk}_{kin mathbb Z^n }$ - the set of grids. For functions $f$ belonging to some functional space $B$ the norm of $L$ in the conjugate space $B*$ is minimizied about coefficients ${c_k}$. The optimal error functional is replaced by asymptotic optimal one with more simple rools for the computation of coefficients. begin{equation} label{n2} exists h_0, exists Cquad forall hin(0,h_0): left(rho(hHk,mathbb R^nsetminusOmega)geq Ch right)Longrightarrow(c_k(hequiv 1)) end{equation} and begin{equation} label{n3} exists C forall h,k qquad |c_k(h)|leq C. end{equation} With say that the formula(ref{n1}) under conditions (ref{n2}),(ref{n3}) has the Limited Boundary Layer and becomes LBL-formula. We propose to change (ref{n2}) on the weakly condition begin{equation} label{n4} exists h_0, exists Cquad forall hin(0,h_0): left(rho(hHk,mathbb R^nsetminusOmega)geq Ch|ln h| right)Longrightarrow(c_k(hequiv 1)) end{equation} Then the cubature formula conserves the asymptotic optimal properties. The more possibility for calculation of coefficients can be applied to get better properties of the formula. For example it can be applied for better approximations of optimal formulas. Because we consider cubature formulas with conditions (ref{n3}),(ref{n4}) and call its the LBL-formulas with logarithmic boundary layer.

Примечание. Тезисы докладов публикуются в авторской редакции

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