Pontryagin space operator valued generalized Schur functions and generalized Nevanlinna functions are investigated by using discrete-time systems, or operator colligations, and state space realizations. It is shown that generalized Schur functions have strong radial limit values almost everywhere on the unit circle. These limit values are contractive with respect to the indefinite inner product, which allows one to generalize the notion of an inner function to Pontryagin space operator valued setting. Transfer functions of self-adjoint systems such that their state spaces are Pontryagin spaces, are generalized Nevanlinna functions, and symmetric generalized Schur functions can be realized as transfer functions of self-adjoint systems with Kreĭn spaces as state spaces. A criterion when a symmetric generalized Schur function is also a generalized Nevanlinna function is given. The criterion involves the negative index of the weak similarity mapping between an optimal minimal realization and its dual. In the special case corresponding to the generalization of an inner function, a concrete model for the weak similarity mapping can be obtained by using the canonical realizations.

中文翻译：

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广义 Schur-Nevanlinna 函数及其实现

Pontryagin 空间算子值广义 Schur 函数和广义 Nevanlinna 函数通过使用离散时间系统或算子关联和状态空间实现进行研究。结果表明，广义 Schur 函数在单位圆上几乎处处都具有很强的径向极限值。这些极限值相对于不定内积是收缩的，这允许人们将内函数的概念推广到 Pontryagin 空间算子值设置。自伴随系统的传递函数，其状态空间是庞特里亚金空间，是广义 Nevanlinna 函数，对称广义 Schur 函数可以实现为自伴随系统的传递函数，以 Kreĭn 空间为状态空间。给出了对称广义 Schur 函数也是广义 Nevanlinna 函数时的判据。该标准涉及最优最小实现与其对偶之间的弱相似映射的负索引。在对应于内函数泛化的特殊情况下，可以通过使用规范实现来获得弱相似性映射的具体模型。