On an inequality of Sidon type for trigonometric polynomials (Q650462)

The author establishes a nice lower bound of the uniform norm of a trigonometric polynomial (with particular frequencies) by the \(\ell^1\)-sum of the \(L^1\)-norm of its summand. More precisely, the following result is proved: For any trigonometric \(f\) of the form \(f(x)=\displaystyle\sum_{k=l+1}^{2l} p_k(x)\cos(n_kx)\) with deg(\(p_k\))\(\leq a(\lambda)n_{l+1}/l\), we have \[ \|f\|_\infty\geq c(\lambda)\displaystyle\sum_{k=l+1}^{2l}\|p_k\|_1, \] where \((n_k)\) is a lacunary sequence with ratio \(\lambda>1\), the constant \(c(\lambda)\) depends on \(\lambda\) only and \(a(\lambda)=\min\big((\lambda-1)/5,1/5\big)\). This generalizes a previous result due to B. S. Kashin and V. N. Temlyakov, where \(n_k=4^k\). Nevertheless, even in this case, the result of the paper under review is sharper (the degree allowed for the \(p_k\)'s may be larger here). As one may expect, the proof relies on a delicate (and technical) use of generalized Riesz products as well as generalized De La Vallée-Poussin kernels.