Norms of Dirichlet kernels and some other trigonometric polynomials in $L_p$-spaces

M. I. Dyachenko

Sbornik. Mathematics, Tome 78 (1994) no. 2, pp. 267-282

Résumé

The following problem is considered. Let $\mathbf{a}=\{a_{\mathbf{n}}\}_{\mathbf{n}=1}^{\mathbf{M}}=\{a_{n_1,\dots,n_m}\}_{n_1,\dots,n_m=1}^{M_1,\dots,M_m}$ be a finite $m$-fold sequence of nonnegative numbers such that if $\mathbf{n}\ge\mathbf{k}$ then $a_{\mathbf{n}}\le a_{\mathbf{k}}$, and $Q(\mathbf{x})=\sum_{\mathbf{n}=1}^{\mathbf{M}}a_{\mathbf{n}}e^{i\mathbf{nx}}$. The purpose of the work is to give best possible upper estimates of the norms $\|Q(\mathbf x)\|_p$ and $\|Q(\mathbf x)\|_{\mathbf{\delta},p}$ with $\boldsymbol\delta>0$ in terms of the coefficients $\{a_{\mathbf{n}}\}$. The Dirichlet kernels $D_U(\mathbf{x})=\sum_{\mathbf{n}\in U}e^{i\mathbf{nx}}$ with $U\in A_1$ present a particular case of $Q(\mathbf x)$.