Estimating the $L_p$-norm of an algebraic polynomial in terms of its values at the nodes of a uniform grid

An estimate of the $L_p$-norm, $p\geqslant 1$, of an arbitrary algebraic polynomial of degree $\leqslant n$ in terms of its values at $N>n$ nodes of a uniform grid is obtained. This estimate shows, in particular, that for $N\geqslant \theta n^2$ with $\theta >0$ the $L_p$-norm of a polynomial grows as $n\to\infty$ not faster than the $L_q$-means, $q\geqslant p$, of this polynomial over the nodes of the grid times some power of $n$.