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

D. S. Lubinsky ()
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D. S. Lubinsky: Georgia Institute of Technology, School of Mathematics

A chapter in Trigonometric Sums and Their Applications, 2020, pp 119-147 from Springer

Abstract: Abstract 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.

Keywords: Marcinkiewicz-Zygmund inequalities; Quadrature sums; Airy functions; Hermite polynomials (search for similar items in EconPapers)
Date: 2020
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-030-37904-9_6

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DOI: 10.1007/978-3-030-37904-9_6

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