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Аппроксимация функций и квадратурные формулы
It is investigated the transcendential equation of unknown function $s$
begin{equation}
label{1}
intlimits_0^frac{1}{2}|B_m(x)|^s sign(B_m(x))dx=0,
end{equation}
where $B_m(x)$ --- the Bernoully polynomial of $m$ degree, $s=q-1,$
$p,qin(1,infty),$ $frac{1}{p}+frac{1}{q}=1.$ Solutions of this
equation define, what sequences of quadrature formulae are asymptotically
optimal in spaces $L_p^m(a,b)$ on dependence from degrees of summing of
derivatives integrable functions.
Number $s$ is a solution of equation (ref{1}) if and only if sequences
of quadrature formulae with regular boundary layer are asymptotically
optimal in $L_p^m(a,b), quad pin(1,infty).$ If $m$ is an odd number
then Sobolev's formulae with regular boundary layer are asymptotically
optimal in $L_p^m(a,b),$ $pin(1,infty),$ $-infty
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Дата последней модификации: 06-Jul-2012 (11:45:20)