The existence of optimal quadrature formulas with given multiplicities of nodes

Suppose that $R_p(\overline x)$ is the error of the best method of integration in the class $W^r_p[a,b]$ with nodes $(x_k)_1^n$ of multiplicities $(\nu_k)_1^n$, i.e. $\overline x=\{(x_1,\nu_1),\dots,(x_n,\nu_n)\}$. It is then shown that for $1<p<\infty$ and for every system of multiplicities $(\nu_k)_1^n$ with $1\leqslant\nu_k\leqslant r$ for $k=1,\dots,n$, the lower bound
$$ \inf\bigl\{R_p(\overline x)\mid\overline x=\{(x_1,\nu_1),\dots,(x_n,\nu_n)\},\,a\leqslant x_1<\dots<x_n\leqslant b\bigr\} $$
is attained for some nodes $(x^*_k)_1^n$ with exactly the multiplicities $(\nu_k)_1^n$. Moreover, $a<x^*_1$ and $x^*_n<b$ .
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