乘积正函数、临界指数现象和Jain-Karlin-Schoenberg核
Multiply positive functions, critical exponent phenomena, and the Jain-Karlin-Schoenberg kernel
摘要：
我们证明了Karlin [Trans. AMS 1964]的一个结果的逆命题，并加强了他的结果和Schoenberg [Ann. of Math.1955]的两个结果.后者的结果之一涉及多重正函数的拉普拉斯变换的零点。另一个研究结果表明，两个特定核的$α$的幂是完全非负的$p\geq 2$阶（表示为TN $_p$）;两位作者都证明了$α\geq p-2$会发生这种情况，Schoenberg证明了$α<p-2$不会。我们更强烈地表明，对于每一个$p \times p$子矩阵的内核，直到一个“移位”，其$α$次幂是完全积极的顺序$p$（TP $_p$）的每一个$α> p-2$，并不是TN $_p$的每一个非整数$α\in(0,p-2)$。特别是，这些结果揭示了“临界指数”的现象，在总的积极性。我们还证明了相反的1968年的结果卡林，揭示了另一个关键的指数现象-拉普拉斯变换的所有波利亚频率（PF）的功能。我们进一步分类的权力保持所有TN $_p$汉克尔内核的间隔，并隔离个别内核编码这些权力。然后我们将Polya-Szego（1925）、Loewner/Horn（1969）和Khare-Tao（出版中）关于正矩阵的结果转移到Hankel TN $_p$核上。另一个应用构造编码Loewner凸幂的各个矩阵。这补充了Jain（2020）关于Loewner积极性的结果，我们将其加强为总积极性。值得注意的是，Jain的这些（强化的）结果，Schoenberg和Karlin的结果，后者的逆结果，以及上述Hankel核都来自一个对称的二阶核及其幂：$\max(1+xy,0)$。我们还提供了一个新的PF功能和序列的顺序$p\geq 3$，勋伯格1951年的结果$p=2$。我们纠正了一个小的差距，在他的论文中，在分类的不连续PF功能。
英文摘要：
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.
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