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
![\lambda_k](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/8ff9c1b69b4201fec1b23780372d5cdf.png)
Under an assumption on the gap
![\gamma_M](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/b0fdc2e1fc6210d8751d55ab3347fb99.png)
![\lambda_k](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/8ff9c1b69b4201fec1b23780372d5cdf.png)
![M\to + \infty](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/b44d16ef0a8dd2c8a24436f0abc36911.png)
![c_M](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/ef4c9e03818e49d75b6944261fb814af.png)
Keywords:
Trigonometric polynomials; inequalities; Parseval equality
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