Fine spectral estimates with applications to the optimally fast solution of large FDE linear systems

Consider the real symbol $f_{n}(\thetat)\coloneqq\sum_{j=0}^{n-1}h^{jh}|\thteta|^{2-jh}$ with $h\coloneqq\frac{1}{n}$. In this note we obtain a real interval where the smallest eigenvalue of the Toeplitz matrix $A_{n}\coloneqq T_{n}(f_{n})$ belongs to, with $f_{n}$ being the standard generating function of $A_{n}$. The considered type of matrix stems in the context of the numerical approximation of distributed order fractional differential equations (FDEs). In fact the new presented bounds improve those already present in the literature and give a more accurate spectral information, which are in fact used in the design of fast numerical algorithms for the associated linear systems, approximating the given distributed order FDEs.