## 2 Spectral Analysis
### Preconditioning  
For a symmetric Toeplitz matrix \(T_{n}\in\mathbb{R}^{n\times n}\) with first column \([t_{1},t_{2},\ldots,t_{n}]^{\top}\), the matrix \(\tau(T_{n})\) defined as  
\[\tau(T_{n}):=T_{n}-H(T_{n}) \tag{9}\]  
the natural \(\tau\) preconditioner of \(T_{n}\) was already considered decades ago in [10; 11; 12] when a great amount of theoretical and computational work was dedicated to preconditioning strategies for structured linear systems. Here, \(H(T_{n})\) denotes a Hankel matrix whose entries are constant along each antidiagonal and whose precise definition is the following: the first row and the last column of \(H(T_{n})\) are given by \([t_{2},t_{3},\ldots,t_{n-1},0,0]\) and \([0,0,t_{n-1},\ldots,t_{3},t_{2}]^{\top}\), respectively. Notice, that by using the sine transform matrix \(S_{n}\), defined as  
\[[S_{n}]_{k,j}=\sqrt{\frac{2}{n+1}}\sin\left(\frac{\pi kj}{n+1}\right),\quad 1 \leq k,j\leq n,\]  
it is known that every \(\tau\) matrix is diagonalized as \(\tau(T_{n})=S_{n}\Lambda_{n}S_{n}\), where \(\Lambda_{n}\) is a diagonal matrix constituted by all eigenvalues of \(\tau(T_{n})\), and \(S_{n}=([S_{n}]_{jk})\) is the real, symmetric, orthogonal matrix defined before, so that \(S_{n}=S_{n}^{\top}=S_{n}^{-1}\). Furthermore, the matrix \(S_{n}\) is associated with the fast sine transform of type I (see [13; 14] for several other sine/cosine transforms). Indeed the multiplication of a matrix \(S_{n}\) times a real vector can be conducted in \(\mathcal{O}(n\log n)\) real operations and the cost is around half of the celebrated discrete fast Fourier transform [15]. Therefore, all the relevant matrix operations in this algebra cost \(\mathcal{O}(n\log n)\) real operations, including matrix-matrix multiplication, inversion, solution of a linear system, and computation of the spectrum, i.e., of the diagonal entries of \(\Lambda_{n}\). Using standard and known techniques, the \(\tau\) algebra has \(d\)-level versions for every \(d\geq 1\), in which a \(\tau(T_{\mathbf{n}})\), \(T_{\mathbf{n}}\)\(d\)-level symmetric Toeplitz matrix of size \(v(\mathbf{n})=n_{1}n_{2}\cdots n_{d}\), \(\mathbf{n}=(n_{1},\ldots,n_{d})\), has the diagonalization form  
\[\tau(T_{\mathbf{n}})=S_{\mathbf{n}}\Lambda_{\mathbf{n}}S_{\mathbf{n}},\ \ \ S_{ \mathbf{n}}=S_{n_{1}}\otimes\cdots\otimes S_{n_{d}}, \tag{10}\]  
with \(\Lambda_{\mathbf{n}}\), the diagonal matrix is obtained as a \(d\)-level sine transform of type I of the first column of \(T_{\mathbf{n}}\). Again the quoted \(d\)-level transform and all the relevant matrix operations in the related algebra have a cost of \(\mathcal{O}(\
u(\mathbf{n})\log\
u(\mathbf{n}))\) real operations which is quasi-optimal given the fact that the matrices have size \(\
u(\mathbf{n})\). At the algebraic level, the explicit construction can be conducted recursively using additive decomposition (9): first at the most external level, and then applying the same operation to any block which is a \((d-1)\)-level symmetric Toeplitz matrix and so on until arriving at matrices with scalars. In light of the excellent structural, spectral, and computation features of the \(\tau\) algebra in \(d\) levels, two different types of \(\tau\) preconditioning for the related linear systems were proposed in [2] and one in [1] (with \(d=2\) and \(\mathbf{n}=(n,n)\)). Here, we consider the latter. In fact, \(\{T_{\mathbf{n}}(f)-\tau(T_{\mathbf{n}}(f))\}_{n}\sim_{\sigma,\lambda}0\) for any Lebesgue integrable \(f\), thanks to the distribution results on multilevel Hankel matrix-sequences generated by any \(L^{1}\) function \(f\) proven in [16]. From this, as proven in [1], the preconditioner \(P_{\mathbf{n}}=\tau(T_{\mathbf{n}}(t_{a}))\) is such that the preconditioned matrix sequence is clustered at \(1\) both in the eigenvalue and singular value sense under mild assumptions. In fact, using the notion of an approximating class of sequences, the eigenvalue perturbation results in [6], and the GLT apparatus [3]; it is enough that \(\mu(x,y)\) is Riemann integrable or simply bounded.