AI 中文总结
该研究构造了满足特定傅里叶不变性与稠密零点条件的函数，否定了Radchenko和Viazovska的相关问题，证明了特定序列为希尔伯特空间的通用插值序列（β>1/2时不成立）。
AI 中文摘要
对每个满足0≤β≤1/2的β，我们构造了一个非零实值连续函数f_β，其属于L¹(ℝ)∩L²(ℝ)，满足傅里叶变换Fourier变换下的性质：$\tilde{f}_β = f_β$，且对所有n≥0，有$f_β(\frac{\root\radic n}{[\text{log}(e+n)]^β}) = 0$。当β=0时，该构造否定了Radchenko与Viazovska提出的关于其傅里叶插值公式的问题。该构造使用由傅里叶不变埃尔米特函数生成的一系列再生核希尔伯特空间，通过Mehler公式确定这些空间的再生核，借助对这些核的恰当估计，证明当添加一个辅助点后，序列$\frac{\root\radic n}{[\text{log}(e+n)]^β}$是所考虑的至少一个希尔伯特空间的通用插值序列，但当β>1/2时该结论不成立。
英文摘要
For every $0\leqβ\leq1/2$, we construct a nonzero real-valued continuous function $f_β$ in $L^1(\mathbb R)\cap L^2(\mathbb R)$ such that $\widehat {f}_β=f_β$ and $f_β(\sqrt{n}/[\log(e+n)]^β)=0$ for all $n\geq 0$. The case $β=0$ settles in the negative a question raised by Radchenko and Viazovska regarding their Fourier interpolation formula. The construction uses a scale of reproducing kernel Hilbert spaces generated by the Fourier-invariant Hermite functions. Applying the Mehler formula, we identify the reproducing kernels of these spaces. By suitable estimates of these kernels, we show that $(\sqrt{n}/[\log(e+n)]^β)$, with one auxiliary point added to it, is a universal interpolating sequence for at least one of the Hilbert spaces under consideration. However, this result fails when $β>1/2$.
