We propose a conceptual frame to interpret the prolate differential operator, which appears in Communication Theory, as an entropy operator; indeed, we write its expectation values as a sum of terms, each subject to an entropy reading by an embedding suggested by Quantum Field Theory. This adds meaning to the classical work by Slepian et al. on the problem of simultaneously concentrating a function and its Fourier transform, in particular to the ``lucky accident" that the truncated Fourier transform commutes with the prolate operator. The key is the notion of entropy of a vector of a complex Hilbert space with respect to a real linear subspace, recently introduced by the author by means of the Tomita-Takesaki modular theory of von Neumann algebras. We consider a generalization of the prolate operator to the higher dimensional case and show that it admits a natural extension commuting with the truncated Fourier transform; this partly generalizes the one-dimensional result by Connes to the effect that there exists a natural selfadjoint extension to the full line commuting with the truncated Fourier transform.
翻译:我们提出一个概念框架来解释波罗拉特差分运算人,它出现在通信理论中,作为一个环球运算人;事实上,我们把它的预期值写成一个术语的总和,每个参数都以Quantum Field理论建议的嵌入式读取为对象。这增加了Slepian等人关于同时集中一个函数及其Fourier变异的经典工作的意义,特别是“幸运的事故”,该事故是,Fourier变异与普罗列特运算人截断的通航。关键在于,一个复杂的Hilbert空间的矢量在真实的线性子空间的变异性,这是作者最近通过von Neuumann代列布拉斯的托米塔-Takesaki模块理论引进的。我们考虑将原形运算操作人与更高维度的立体变异体进行一般化,并表明它承认自然延伸与Tyier变形变异的波变形连接;这在一定程度上概括了Connes的一维结果,其效果是存在与四面变形变形的全线的自然自连接延伸延伸延伸延伸延伸扩展。</s>