Ingham type inequalities towards Parseval equality
Abstract
We consider Trigonometric series with real exponents 
: 
between , we show the inequality \begin{equation*}\label{conf000} \frac {2\pi}{\gamma_M(2-c_M)}\sum_{n=1}^M\vert x_n\vert^2 \leq \int_{-\pi/\gamma_M}^{\pi/\gamma_M}\vert \sum_{k=1}^{M} x_ke^{i \lambda_kt}\vert^2dt\leq \frac {2\pi}{c_M\gamma_M} \sum_{n=1}^M\vert x_n\vert^2 \end{equation*} and we show for a class of problems that the limit as 
leads to the Parseval's equality. The role of constants 
in the above formula is one of the key points of the paper
: Under an assumption on the gap
between , we show the inequality \begin{equation*}\label{conf000} \frac {2\pi}{\gamma_M(2-c_M)}\sum_{n=1}^M\vert x_n\vert^2 \leq \int_{-\pi/\gamma_M}^{\pi/\gamma_M}\vert \sum_{k=1}^{M} x_ke^{i \lambda_kt}\vert^2dt\leq \frac {2\pi}{c_M\gamma_M} \sum_{n=1}^M\vert x_n\vert^2 \end{equation*} and we show for a class of problems that the limit as leads to the Parseval's equality. The role of constants in the above formula is one of the key points of the paperKeywords:
Trigonometric polynomials; inequalities; Parseval equality
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