Uniform bounds on the 1-norm of the inverse of lower triangular Toeplitz matrices
Consider the lower triangular \(\left( n+1\right) \times \left( n+1\right) \) Toeplitz matrix \[ T_{n}=\left( \begin{matrix} b_{0} & 0 & \cdots & 0 \\ b_{1} & b_{0} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ b_{n} & b_{n-1} & \cdots & b_{0} \end{matrix} \right) \] where \(b_{0}\geq b_{1}\geq \cdots \geq b_{n}\geq b\geq 0\) and \( \lim_{n\rightarrow \infty }b_{n}=b=0\). The authors provide a sharp uniform upper bound for \(\left\| T_{n}^{-1}\right\| _{\infty }\). In particular case where \[ b_{0}=\frac{1}{\sqrt{2}},b_{1}=\frac{\sqrt{3}-1}{\sqrt{2}},b_{2}=\frac{\sqrt{ 5}-\sqrt{3}}{\sqrt{2}},\dots,b_{n}=\frac{\sqrt{2n+1}-\sqrt{2n-1}}{\sqrt{2}} \] then it is proven that \[ \left\| T_{n}^{-1}\right\| _{1}\leq 2\sqrt{2}\left( 5-\sqrt{3}-\sqrt{5} \right) \] and thus a solution is given to the GKM conjecture. Furthermore, in case where \[ b_{0}=1,b_{1}=2^{1-a}-1,b_{2}=3^{1-a}-2^{1-a},\dots,b_{n}=\left( n+1\right) ^{1-a}-n^{1-a} \] with \(a\in \left( 0,1\right) \), upper bounds are given for \(\left\| T_{n}^{-1}\right\| _{1}\) under certain conditions, by providing in this way a partial answer to the Brunner's one-point collocation problem.
inverse of lower triangular Toeplitz matrix
scientific article; zbMATH DE number 5924527