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On Marcinkiewicz-Zygmund inequalities at Hermite zeros and their Airy function cousins. (English) Zbl 1446.41006

Raigorodskii, Andrei M. (ed.) et al., Trigonometric sums and their applications. Cham: Springer. 119-147 (2020).
Let \(a_j\), \(j \ge 1\), be the zeros of the Airy function \(\mathrm{Ai}\). Under appropriate conditions on the function \(f\), the author proves forward Marcinkiewicz-Zygmund inequalities of the form \(\int_{-\infty}^\infty |f(t)|^p \, \mathrm d t \ge A \frac{\pi^2}6 \sum_{k=1}^\infty \frac{|f(a_k)|^p}{\mathrm{Ai}'(a_k)^2}\) for \(p \in [1, \infty]\) and the corresponding converse inequalities \(\int_{-\infty}^\infty |f(t)|^p \, \mathrm d t \le B \frac{\pi^2}6 \sum_{k=1}^\infty \frac{|f(a_k)|^p}{\mathrm{Ai}'(a_k)^2}\) for \(p \in (1,4)\). The best possible constants \(A\) and \(B\), respectively, in these inequalities are discussed in detail.
For the entire collection see [Zbl 1443.39001].

MSC:

41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
65D30 Numerical integration
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