On the 𝐿² inequalities involving trigonometric polynomials and their derivatives

Chen, Weiyu

doi:10.1090/S0002-9947-1995-1254834-7

On the $L^ 2$ inequalities involving trigonometric polynomials and their derivatives
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Trans. Amer. Math. Soc. 347 (1995), 1753-1761 Request permission

Abstract:

In this note we study the upper bound of the integral \[ \int _0^\pi {{{({t^{(k)}}(x))}^2}w(x)} dx\] where $t(x)$ is a trigonometric polynomial with real coefficients such that $\left \| t \right \|\infty \leqslant 1$ and $w(x)$ is a nonnegative function defined on $[0,\pi ]$. When $w(x) = \sin ^jx$, where $j$ is a positive integer, we obtain the exact upper bound for the above integral.

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