On Sharp Constants in Marcinkiewicz-zygmund and Plancherel-polya Inequalities

The Plancherel-Polya inequalities assert that for 1 < p <∞, and entire functions f of exponential type at most π, Ap ∞ ∑ j=−∞ |f (j)| ≤ ∫ ∞ −∞ |f | ≤ Bp ∞ ∑ j=−∞ |f (j)| . The Marcinkiewicz-Zygmund inequalities assert that for n ≥ 1, and polynomials P of degree ≤ n− 1, Ap n n ∑ j=1 ∣∣∣P (e2πij/n)∣∣∣p ≤ ∫ 1 0 ∣∣P (e2πit)∣∣p dt ≤ B′ p n n ∑ j=1 ∣∣∣P (e2πij/n)∣∣∣p . We show that the sharp constants in both inequalities are the same, that is Ap = Ap and Bp = B ′ p. Moreover, the two inequalities are equivalent. We also discuss the case p ≤ 1.