From Matrices to Operators: A Tensorial View on the Legendre Tau Method for Time-Delay Systems

It is the essence of numerical methods like the Legendre tau method to approximate infinite-dimensional problems by finite-dimensional ones. Instead of functions and operators, finite-dimensional vectors and matrices occur. However, such a vector or matrix is still the coordinate representation of a function or operator that approximates the original one. The present paper considers the coordinate representations that result from the Legendre tau method for time-delay systems. A combination of two different interpretations of the coordinates turns out to be particularly helpful: a coordinate vector can be seen as representing a polynomial of degree $N$ or as representing a polynomial of degree $N-1$ and a discontinuous end point. The fact that the associated basis functions are nonorthonormal in $L_2 \times \mathbb R^n$ must be dealt with. Tools from tensor algebra are used to make explicit the operators that are obtained as matrix representations from the Legendre tau method.