Trigonometric series with positive partial sums (Q1326062)

In the present paper a trigonometric series with the spectrum \(A\) is considered, \(\sum_{\nu\in A} c_ \nu e^{i\nu x}\), \(A\subset\mathbb{Z}^ n\), \(c_ \nu= \overline{c_{-\nu}}\). By \(B^*= \{z: z= x-y, z\neq 0, x,y\in B\}\) the difference set of \(B\) is denoted. Let \(f(x)\) be a continuous function on \(T^ n= (-\pi,\pi]^ n\) with zero mean and \(M(f)= \min\{\max_{x\in T^ n} f(x); -\min_{x\in T^ n} f(x)\}\). The main result is the following Theorem 1. Let \(A\subset \mathbb{Z}^ n\) be finite, \(B\subset A\), \(B^*\cap A= \emptyset\), \(t(x)= \sum_{\nu\in A} \widehat{t_ \nu} e^{i\nu x}\), \(\widehat t_ \nu=\overline{\widehat t}_{-\nu}\), \(\widehat t_ 0= 0\). Then \(M(f)\geq \Bigl\{\sum_{\nu\in B} |\widehat t_ \nu|^ 2\Bigr\}^{1/2}\).