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Let $f : \mathbb{R} / \mathbb{Z} \to \mathbb{C}$ be a trigonometric polynomial of degree $n$ and $m-1 \geq n$ be an integer. The Marcinkiewicz-Zygmund inequality asserts $$\int |f|^p \leq \frac{C_p}{m} \sum_{j=1}^m |f(j/m)|^p , \ \ \ 1 < p < \infty.$$

My question is known about the behavior $C_p$, as a function of $p$? I am particularly interested in the case $1 < p < 2$.

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D. Lubinsky in this paper shows that the best constant in the Marcinkiewicz-Zygmund inequality is equal to the best constant in the Polya-Plancherel inequality. This observation and an upper bound for the constant in R. P. Boas' book Entire Functions (Theorem 6.7.15 pg. 101) for the latter inequality may be useful. It appears the best constant is not known precisely ( see comments in Lubinsky's paper, page 2).

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