论文标题
跟踪minmax功能和自由基laguerre-pólya类
Trace minmax functions and the radical Laguerre-Pólya class
论文作者
论文摘要
我们对函数进行分类$ f：（a，b）\ rightarrow \ mathbb {r} $，它满足不等式$$ \ perperatorName {tr} f（a）+f（a）+f（c）\ geq \ geq \ geq \ geq \ geq \ peratatorName {tr} tr} f（b）+f（b）+f（d）$所谓的痕量minmax函数。 （这里$ a \ leq b $如果$ b-a $是正半段的，并且$ f $是通过功能性演算评估的。）当且仅当其派生派衍生物分析地继续前进到上半平面的自我映射时，函数是跟踪minmax的。跟踪minmax函数的负指数$ g = e^{ - f} $满足不平等 $$ \ det g（a）\ det g（c）\ leq \ det g（b）\ det g（d）$$ for $ a，b，c，d $如上所述。我们称此类功能决定性等级。我们表明，决定性等级函数在Laguerre-Pólya类的“根本”中。我们得出了此类功能的积分表示，这实际上是Laguerre-Pólya类功能的Hadamard分解的连续版本。我们应用结果来提供一些等效的Riemann假设。
We classify functions $f:(a,b)\rightarrow \mathbb{R}$ which satisfy the inequality $$\operatorname{tr} f(A)+f(C)\geq \operatorname{tr} f(B)+f(D)$$ when $A\leq B\leq C$ are self-adjoint matrices, $D= A+C-B$, the so-called trace minmax functions. (Here $A\leq B$ if $B-A$ is positive semidefinite, and $f$ is evaluated via the functional calculus.) A function is trace minmax if and only if its derivative analytically continues to a self map of the upper half plane. The negative exponential of a trace minmax function $g=e^{-f}$ satisfies the inequality $$\det g(A) \det g(C)\leq \det g(B) \det g(D)$$ for $A, B, C, D$ as above. We call such functions determinant isoperimetric. We show that determinant isoperimetric functions are in the "radical" of the the Laguerre-Pólya class. We derive an integral representation for such functions which is essentially a continuous version of the Hadamard factorization for functions in the the Laguerre-Pólya class. We apply our results to give some equivalent formulations of the Riemann hypothesis.
