An elementary note on asymptotic properties of Toeplitz and multilevel Toeplitz matrices Original Research Article

Let {aℓ(f)}ℓ=−∞∞ be the Fourier coefficients of fset membership, variantL[−π,π] and consider the Toeplitz matrices {Tn(f)}, where Tn(f)=(ai−j(f))i,j=1n; thus, {Tn(f)} is generated by the symbol f. There are many results on asymptotic properties of {Tn(f)} as n→∞. If r is a positive integer and Tn(r)(f)=(ar(i−j))i,j=1n, then {Tn(r)(f)}={Tn(fr)} where fr is easily obtained from f. Hence, known results on the asymptotic behavior of Toeplitz matrices generated by a symbol can be applied to {Tn(r)(f)}. Although this is elementary, to our knowledge it has not been previously exploited. We extend this idea to multilevel Toeplitz matrices.