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Date
2023-05-29Type
- Working Paper
ETH Bibliography
yes
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Abstract
We prove a conjecture by Vemuri by proving sharp bounds on $\ell^κ$ sums of Hermite functions multiplied by an exponentially decaying factor. More explicitly, we prove that, for each $y>0,$ we have \[ \sum_{n \ge 1} |h_n(x)|^κ \frac{e^{-κn y}}{n^β} \ll_y x^{\frac{1}{2} - 2β} e^{-κx^2 \tanh(y)/2}, \] for all $x \in \mathbb{R}$ sufficiently large. Our proof involves the classical Plancherel-Rotach asymptotic formula for Hermite polynomials and a careful local analysis near the maximum point of such a bound. Show more
Publication status
publishedJournal / series
arXivPages / Article No.
Publisher
Cornell UniversityEdition / version
v1Subject
Classical Analysis and ODEs (math.CA); FOS: MathematicsOrganisational unit
09565 - Figalli, Alessio / Figalli, Alessio
Funding
721675 - Regularity and Stability in Partial Differential Equations (EC)
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ETH Bibliography
yes
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