# A Graduated Non-Convexity Technique for Dealing Large Point Spread Functions
## 4 E-CATILED Technique
### Approximation of Symmetric Toeplitz Matrices
For every \(n\in\mathbb{N}\), set  
\[G_{n}(T_{n})=C_{n}(T_{n})+B_{n}(T_{n})=\min_{G\in\widehat{G}_{n}}\|G-T_{n}\|_{ F}=\min_{G\in\widehat{G}_{n}}\|G-T_{n}\|_{F}, \tag{14}\]1 \[G_{n}(T_{n})=C_{n}(T_{n})+B_{n}(T_{n})=\min_{G\in\widehat{G}_{n}}\|G-T_{n}\|_{ F}=\min_{G\in\widehat{G}_{n}}\|G-T_{n}\|_{F}, \tag{14}\]2Observe that any function \(f\) defined by a power series as in the first line of (17) is real-valued, and the set of such functions satisfying the condition \(\sum_{j=-\infty}^{+\infty}|t_{j}|<+\infty\) is called _Wiener class_ (see, e.g., [19]). Given a function \(f\) belonging to the Wiener class and a matrix \(T_{n}(f)=(t_{kj})_{kj}\in\widehat{\mathcal{T}}_{n}\) such that \(t_{kj,j}=t_{[k-j]},k,j\in\{0,1,\ldots,n-1\}\), and \(f(z)=\sum_{j=-\infty}^{+\infty}t_{j}z^{j}\), then we say that \(T_{n}(f)\) is _generated by \(f\)_. **Theorem 7**. : _For \(n\in\mathbb{N}\), given \(T_{n}(f)\in\widehat{\mathcal{T}}_{n}\), let \(C_{n}(f)=C_{n}(T_{n}(f))\), \(B_{n}(f)=B_{n}(T_{n}(f))\), \(F_{n}(f)=F_{n}(T_{n}(f))\) be as in Theorems 5 and 6, and set \(V_{n}(f)=C_{n}(f)+B_{n}(f)+F_{n}(f)\), and \(G_{n}(f)=C_{n}(f)+B_{n}(f)\). Then, the following statements hold._  
_(i)_ _For every_ \(\varepsilon>0\)_, there is a positive integer_ \(n_{0}\)_, such that for each_ \(n\geq n_{0}\) _and for every eigenvalue_ \(\lambda_{j}^{(V_{n}(f))}\) _of_ \(V_{n}(f)\)_, it is_  
\[G_{n}(T_{n})=C_{n}(T_{n})+B_{n}(T_{n})=\min_{G\in\widehat{G}_{n}}\|G-T_{n}\|_{ F}=\min_{G\in\widehat{G}_{n}}\|G-T_{n}\|_{F}, \tag{14}\]3  
_where_ \(f_{\min}\) _and_ \(f_{\max}\) _denote the minimum and the maximum value of_ \(f\)_, respectively._ _(ii)_ _For every_ \(\varepsilon>0\)_, there is a positive integer_ \(n_{0}\)_, such that for each_ \(n\geq n_{0}\) _and for every eigenvalue_ \(\lambda_{j}^{(G_{n}(f))}\) _of_ \(G_{n}(f)\)_, it is_  
\[G_{n}(T_{n})=C_{n}(T_{n})+B_{n}(T_{n})=\min_{G\in\widehat{G}_{n}}\|G-T_{n}\|_{ F}=\min_{G\in\widehat{G}_{n}}\|G-T_{n}\|_{F}, \tag{14}\]4  
_where_ \(f_{\min}\) _and_ \(f_{\max}\) _denote the minimum and the maximum value of_ \(f\)_, respectively._ _(iii)_ _If_ \(V_{n}(f)\) _is invertible, then for every_ \(\varepsilon>0\)_, there are_ \(k\)_,_ \(n_{1}\in\mathbb{N}\) _such that for each_ \(n\geq n_{1}\)_, the number of eigenvalues_ \(\lambda_{j}^{((V_{n}(f))^{-1}T_{n}(f))}\) _of_ \(V_{n}^{-1}(f)\)__\(T_{n}(f)\) _such that_ \(|\lambda_{j}^{((V_{n}(f))^{-1}T_{n}(f))}-1|>\varepsilon\) _is less than_ \(k\)_, namely, the spectrum of_ \((V_{n}(f))^{-1}\)__\(T_{n}(f)\) _is clustered around_ \(1\)_._ _(iv)_ _If_ \(G_{n}(f)\) _is invertible, then for every_ \(\varepsilon>0\) _there are_ \(k\)_,_ \(n_{1}\in\mathbb{N}\) _such that for each_ \(n\geq n_{1}\) _the number of eigenvalues_ \(\lambda_{j}^{((G_{n}(f))^{-1}T_{n}(f))}\) _of_ \(G_{n}^{-1}(f)\)__\(T_{n}(f)\) _such that_ \(|\lambda_{j}^{((G_{n}(f))^{-1}T_{n}(f))}-1|>\varepsilon\) _is less than_ \(k\)_, namely, the spectrum of_ \((G_{n}(f))^{-1}\)__\(T_{n}(f)\) _is clustered around_ \(1\)_._  
**Proof. ** For \((i)\) and \((iii)\), see [49]; for \((ii)\) and \((iv)\), see [45]. \(\square\)  
This result confirms how both \(\beta\)-matrices and \(\gamma\)-matrices can approximate symmetrical Toeplitz matrices well.