Marcinkiewicz-Zygmund type inequalities for irregular knots and mixed metrics (Q2719491)

The equivalence of the \(L_p\)-norm, where \(1<p<+\infty\), of a real-valued trigonometric polynomial of order at most \(n\) and its discrete one, which corresponds to a uniform grid, was established by J. Marcinkiewicz and A. Zygmund. In this paper the authors prove the Marcinkiewicz-Zygmund type inequalities on the equivalence of a continuous norm of a real-valued trigonometric polynomial of \(l\) variables and its discrete one in the general case of mixed \(L_{\overline p}\)-metrics, where \(\overline p = (p_1,\dots,p_l)\), \(0<p_i\leq +\infty\), \(i=1,\dots,l\), and of non-uniform grids. A new representation formula for a trigonometric polynomial, which contains a parameter, is used to prove the main results.