# Properties of the Toeplitz matrices formed by square-summable sequence (as opposed to absolutely summable)

I've been reading a wonderful monograph by Robert Gray on the Toeplitz and circulant matrices and am curious about the assumption (4.3) of absolute summability of the sequences $\{t_k\}$ that form the sequences of Toeplitz matrices throughout his study of convergence of said sequences of Toeplitz matrices to sequences of circulant matrices. He addresses it in the first paragraph of page 39, by stating that the assumption ensures the existence of the Fourier series defined by the discrete Fourier transform of the sequence $\{t_k\}$.

I am wondering if the existence of the Fourier series is the only reason why this restriction on $\{t_k\}$ is made. That is, would the sequences of Toeplitz matrices formed by square-summable $\{t_k\}$ still converge to sequences of circulant matrices if one can show the existence of a Fourier series defined by the Fourier transform of $\{t_k\}$ (i.e. would Lemma 4.6 hold)?

I am pretty sure it would, however, I am not very confident in my knowledge of Fourier analysis and would appreciate any help.

The reason I am looking at this is that I am dealing with sequences of covariance matrices in my research, including ones that are not absolutely-integrable but are square-integrable and which have a well-defined Fourier transform (s.t. $\operatorname{sinc}(\cdot)$/rectangular function pairing), and am wondering if I can use nice properties of circulant matrices (such as the inverses being circulant) in my asymptotic analysis of the covariance matrices.

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