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：我们基于算子束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视角。作为首个主要结果，我们证明：当选用平移勒让德基函数时，该方法在频域中等价于Padé逼近。研究表明，Lanczos tau方法能够直接生成稀疏且具有自嵌套特性的离散化格式。进一步论证了该方法与伪谱配点法的等价性，其中非零配点选取为正交多项式的零点。此类选择的要义体现在$H^2$范数逼近中：在温和条件下可观测到超几何收敛，且在特殊情形下证明了超收敛性；这两种收敛速度均显著快于先前文献中报道的代数收敛速度。