A Toeplitz matrix $T$ is characterized by $T_{jk}=t_{j-k}$ for some numbers $t_k$ (so its entries are constant along each diagonal). A Toeplitz operator is (for my purposes) a mapping from $\ell^p(\mathbb{Z})$ to $\ell^q(\mathbb{Z})$ which can be represented by a doubly infinite Toeplitz matrix. (Here $q$ could be anything, not necessarily the Holder conjugate of $p$.)
I've seen in the literature, for example at http://jfi.uchicago.edu/~leop/SciencePapers/jstat9_05_p05012.pdf that the eigenvectors of Hermitian Toeplitz operators are given by sinusoids. Similarly, given suitable decay, large but finite Hermitian Toeplitz matrices have approximately sinusoidal eigenvectors. I can look into details about the convergence myself (especially since this is apparently relatively technical), but the issue about the operators seems to be treated as rather trivial in the literature. For example that paper pretty much says it follows immediately from translation invariance, and I can't seem to follow that.
What is the basic idea here? I don't need all the details, I can iron some of them out myself.
If it is relevant, the particular case I am handling has $t_k=a^{|k|}$ where $0<a<1$.
It seems that the intuition is about a connection to circulant matrices. Given a symmetric $n \times n$ Toeplitz matrix $T$, we can define a "circulant approximation" $C$ by $c_{jk}=\begin{cases} t_0 & j=k \\ t_{j-k} + t_{(n-1)-(j-k)} & j \neq k \end{cases}$.
If we now consider the sequence $T_n$ of such Toeplitz matrices generated by a $t \in \ell^1$, and a corresponding sequence of circulant matrices $C_n$, then $T_n$ and $C_n$ approach one another in Hilbert-Schmidt norm. This ensures that the eigenvalues are close.
Additionally, the eigenvectors of the circulant matrix are complex exponentials (specifically, the columns of the discrete Fourier transform, or maybe its inverse). So we might hope that the eigenvectors are close too. How can we understand this similarity?