Use of complex analysis for deriving lower bounds for trigonometric polynomials

Abstract

It is shown that for any distinct natural numbersk ₁,...,k _n and arbitrary real numbersa ₁,...,a _n the following inequality holds:

$$ - \mathop {\min }\limits_x {\mathbf{ }}\sum\limits_{j = 1}^n {a_j } (\cos (k_j x)) - \sin (k_j x)) \geqslant B\left( {\frac{1}{{1 + ln{\mathbf{ }}n}}\sum\limits_{j = 1}^n {a_j^2 } } \right)^{1/2} ,{\mathbf{ }}n \in {\mathbf{ }}\mathbb{N},$$

whereB is a positive absolute constant (for example,B=1/8). An example shows that in this inequality the order with respect ton, i.e., the factor (1 + lnn)^−1/2, cannot be improved. A more elegant analog of Pichorides' inequality and some other lower bounds for trigonometric sums have been obtained.