On Marcinkiewicz-Zygmund Inequalities at Hermite Zeros and Their Airy Function Cousins

We establish forward and converse Marcinkiewicz-Zygmund Inequalities at the zeros a j j ≥ 1 $$\left \{ a_{j}\right \} _{j\geq 1}$$ of the Airy function A i x $$Ai\left ( x\right ) $$ , such as A π 2 6 ∑ k = 1 ∞ f a k p A i ′ a k 2 ≤ ∫ − ∞ ∞ f t p d t ≤ B π 2 6 ∑ k = 1 ∞ f a k p A i ′ a k 2 $$\displaystyle A\frac {\pi ^{2}}{6}\sum _{k=1}^{\infty }\frac {\left \vert f\left ( a_{k}\right ) \right \vert ^{p}}{Ai^{\prime }\left ( a_{k}\right ) ^{2}}\leq \int _{-\infty }^{\infty }\left \vert f\left ( t\right ) \right \vert ^{p}dt\leq B\frac {\pi ^{2} }{6}\sum _{k=1}^{\infty }\frac {\left \vert f\left ( a_{k}\right ) \right \vert ^{p}}{Ai^{\prime }\left ( a_{k}\right ) ^{2}} $$ under appropriate conditions on the entire function f and p. The constants A and B are those appearing in Marcinkiewicz-Zygmund inequalities at zeros of Hermite polynomials. Scaling limits are used to pass from the latter to the former.