since (\[3.2\]) implies that $c_i$’s are not equal, and from (\[4.6\]) we get $$\lambda_0 = c_0 + 2 \sum_{j = 1}^r c_j = 1 \neq 0.$$ Theorem \[thmGKPS\] suggests how to get the nonsingular embedding of $T_n^d$. First, $T_n^d$ should be embedded into the Töplitz matrix $T_p^d$, of order $p - d$, where $p \geq n$ is chosen so that $m = p - d + r$ is a prime number. Then, $T_p^d$ is embedded into the circulant $C_m^d$.

The other possibility is to embed $T_n^d$ into the smallest circulant matrix $C_m^d$, as in (\[4.1\]), and calculate its eigenvalues from (\[4.6\]), in hope that $C_m^d$ is nonsingular. In this case, extensive numerical testing suggests that $C_m^d$ is always positive definite, but we have not been able to prove it.

There are also several other possible embeddings that guarantee the positive semidefiniteness of the circulant matrix $C$.

The first one, constructed by Dembo, Mallows and Shepp in [@Dembo-Mallows-Shepp-89], ensures that the positive definite Töplitz matrix $T$, of order $n$, can be embedded in the positive semidefinite circulant $C$, of order $m$, where $$m \geq 2 \left(n + \kappa_2(T) \frac{n^2}{\sqrt{6}} \right).
  \label{4.7}$$ A few years later, Newsam and Dietrich [@Newsam-Dietrich-94] reduced the 