论文标题
乘以积极功能，关键指数现象和Jain-Karlin-Schoenberg内核
Multiply positive functions, critical exponent phenomena, and the Jain-Karlin-Schoenberg kernel
论文作者
论文摘要
我们证明了与Karlin [Trans。 AMS 1964]，还加强了他的结果和Schoenberg的两个结果[Ann。数学。 1955年]。后者的结果之一是涉及拉普拉斯变换倍数正函数的零。为两个特定核的$α$驱动的其他结果研究完全不为订单$ p \ geq 2 $（表示为tn $ _p $）；两位作者都表明，这种情况以$α\ geq P-2 $而发生，而Schoenberg证明它不适合$α<P-2 $。我们更强烈地表明，每任要点的$ p \ times p $ p $ cobsatrix，最多可移动，它的$α$ th功率完全是$ p $ p $ p $（tp $ _p $）的订单$α> p-2 $的，而不是tn $ _p $，并且对于每个非Integer $α\ in（0，p-2）。特别是，这些结果揭示了总阳性的“关键指数”现象。我们还证明了与1968年的Karlin结果相反，这揭示了另一个关键指数现象 - 对于所有Polya频率（PF）函数的拉普拉斯变换。 我们进一步对保留所有TN $ _P $ HANKEL内核的功率进行分类，并隔离编码这些功能的单个内核。然后，我们通过Polya-Szego（1925），Loewner/Horn（1969）和Khare-Tao（印刷中）将结果转移到保存器上，从正矩阵到Hankel tn $ _p $ kernels。另一个应用程序构建了编码Loewner凸功率的单个矩阵。这补充了Ja那教的结果（2020年），以提高Loewner的阳性，我们可以增强其总积极性。值得注意的是，这些（加强）Ja那教的结果，Schoenberg和Karlin的结果，后者的匡威和上述Hankel内核都来自单个对称的级别-Two-Two-two-two-two-two-two-two-two-two-two-two：$ \ max（1+xy，0）$。 我们还提供了Schoenberg的1951年结果$ P = 2 $的PF函数和顺序$ P \ GEQ 3 $序列的新颖表征。在不连续的PF函数分类中，我们在他的论文中纠正了一个小差距。
We prove the converse to a result of Karlin [Trans. AMS 1964], and also strengthen his result and two results of Schoenberg [Ann. of Math. 1955]. One of the latter results concerns zeros of Laplace transforms of multiply positive functions. The other results study which powers $α$ of two specific kernels are totally non-negative of order $p\geq 2$ (denoted TN$_p$); both authors showed this happens for $α\geq p-2$, and Schoenberg proved that it does not for $α<p-2$. We show more strongly that for every $p \times p$ submatrix of either kernel, up to a 'shift', its $α$th power is totally positive of order $p$ (TP$_p$) for every $α> p-2$, and is not TN$_p$ for every non-integer $α\in(0,p-2)$. In particular, these results reveal 'critical exponent' phenomena in total positivity. We also prove the converse to a 1968 result of Karlin, revealing yet another critical exponent phenomenon - for Laplace transforms of all Polya Frequency (PF) functions. We further classify the powers preserving all TN$_p$ Hankel kernels on intervals, and isolate individual kernels encoding these powers. We then transfer results on preservers by Polya-Szego (1925), Loewner/Horn (1969), and Khare-Tao (in press), from positive matrices to Hankel TN$_p$ kernels. Another application constructs individual matrices encoding the Loewner convex powers. This complements Jain's results (2020) for Loewner positivity, which we strengthen to total positivity. Remarkably, these (strengthened) results of Jain, those of Schoenberg and Karlin, the latter's converse, and the above Hankel kernels all arise from a single symmetric rank-two kernel and its powers: $\max(1+xy,0)$. We also provide a novel characterization of PF functions and sequences of order $p\geq 3$, following Schoenberg's 1951 result for $p=2$. We correct a small gap in his paper, in the classification of discontinuous PF functions.
