We reformulate the Lanczos tau method for the discretization of time-delay systems in terms of a pencil of operators, allowing for new insights into this approach. As a first main result, we show that, for the choice of a shifted Legendre basis, this method is equivalent to Pad\'e approximation in the frequency domain. We illustrate that Lanczos tau methods straightforwardly give rise to sparse, self nesting discretizations. Equivalence is also demonstrated with pseudospectral collocation, where the non-zero collocation points are chosen as the zeroes of orthogonal polynomials. The importance of such a choice manifests itself in the approximation of the $H^2$-norm, where, under mild conditions, super-geometric convergence is observed and, for a special case, super convergence is proved; both significantly faster than the algebraic convergence reported in previous work.

翻译：我们以算子铅笔的形式重新表述了用于时滞系统离散化的Lanczos tau方法，从而为该方法提供了新的理解。作为第一个主要结果，我们证明在选取平移Legendre基的情况下，该方法等价于频域中的Padé逼近。我们阐释了Lanczos tau方法可直接导出稀疏且具有自嵌套特性的离散格式。该方法也与伪谱配点法等价，其中非零配点被选为正交多项式的零点。这种选择的重要性体现在$H^2$范数的逼近中：在温和条件下可观测到超几何收敛性，而在特殊情况下可证明其具有超收敛性；这两种收敛速度均显著快于以往文献中报告的代数收敛速度。