Toeplitz-to-Circulant Aliasing Identity

• The Toeplitz-to-Circulant Aliasing Identity is a set of exact and asymptotic relations that recast nonperiodic Toeplitz convolution into periodic or anti-periodic circulant convolution while explicitly controlling boundary aliasing.
• It underpins efficient numerical methods by facilitating FFT diagonalization and diagonalizing circulant or skew-circulant components, which are key to preconditioning, Sylvester solvers, and multigrid techniques.
• The identity reduces computational complexity in eigen-decomposition and system inversion by converting Toeplitz structure into forms that allow rapid, FFT-based operator analysis and optimal circulant approximations.

The Toeplitz-to-Circulant Aliasing Identity is the family of exact and asymptotic relations that convert the nonperiodic, linear-convolution structure of a Toeplitz matrix into periodic or anti-periodic convolution encoded by circulant-type matrices. In its most basic form, Toeplitz coefficients are folded modulo the matrix size, so wrap-around images such as $t_{k+mn}$ or $a_{i-j+\ell N}$ become circulant coefficients; in exact finite-dimensional formulations, the wrap-around contribution is either isolated as a skew-circulant or Hankel-like term, or eliminated by embedding into a larger circulant. These identities are central to FFT diagonalization, circulant and block-circulant preconditioning, CSCS iterations for Sylvester equations, multigrid coarse-grid symbol formation, QFT block-encoding on bounded domains, and several operator-theoretic decompositions used in random matrix theory (Liu et al., 2021, Zhu et al., 2016, Javanmard et al., 16 May 2026).

1. Exact finite-dimensional identities

For a Toeplitz matrix $A \in \mathbb{C}^{n\times n}$ with entries $A_{j,k}=a_{j-k}$, one exact formulation is the circulant-and-skew-circulant splitting

$A = A_c + A_s.$

In the CSCS framework, the splitting is implemented coefficient-wise through the first column and first row. If $c=(a_0,a_1,\dots,a_{n-1})^T$ is the first column and $r=(a_0,a_{-1},\dots,a_{-(n-1)})$ the first row, then Algorithm 2 constructs

$$\mathrm{CAc} = \frac{c + [0,\operatorname{reverse}(r(2\!:\!n))]}{2}, \qquad \mathrm{CAr} = [r(1)/2,\operatorname{reverse}(\mathrm{CAc}(2\!:\!n))],$$

and

$$\mathrm{SAc} = \frac{c - [0,\operatorname{reverse}(r(2\!:\!n))]}{2}, \qquad \mathrm{SAr} = [r(1)/2,-\operatorname{reverse}(\mathrm{SAc}(2\!:\!n))].$$

Then $$A_c=\operatorname{toeplitz}(\mathrm{CAc},\mathrm{CAr})$$ and $a_{i-j+\ell N}$0, giving $a_{i-j+\ell N}$1 exactly. The “reverse-and-pad-0” operation is the finite-dimensional implementation of the wrap-around indices $a_{i-j+\ell N}$2 (Liu et al., 2021).

This exactness is not confined to the CSCS notation. A separate formulation writes a Toeplitz matrix $a_{i-j+\ell N}$3 as

$a_{i-j+\ell N}$4

where $a_{i-j+\ell N}$5 is circulant and $a_{i-j+\ell N}$6 is skew-circulant, provided the sequences satisfy

$a_{i-j+\ell N}$7

Entrywise, positive and negative Toeplitz lags are recovered by modulo-$a_{i-j+\ell N}$8 indexing in the circulant term plus the sign-corrected wrap-around in the skew-circulant term (Bose et al., 15 May 2026).

A third exact form is the larger circulant embedding. For Hermitian Toeplitz $a_{i-j+\ell N}$9, if $A \in \mathbb{C}^{n\times n}$0 and $A \in \mathbb{C}^{n\times n}$1 is the $A \in \mathbb{C}^{n\times n}$2 circulant built from

$A \in \mathbb{C}^{n\times n}$3

then

$A \in \mathbb{C}^{n\times n}$4

Here there is no aliasing correction term because the embedding size is large enough to reproduce linear convolution exactly (Zhu et al., 2016).

2. Spectral diagonalization and convolutional meaning

The operational force of the identity comes from the fact that circulant and skew-circulant components are diagonalizable by Fourier transforms. For the unitary DFT matrix $A \in \mathbb{C}^{n\times n}$5 of order $A \in \mathbb{C}^{n\times n}$6,

$A \in \mathbb{C}^{n\times n}$7

where $A \in \mathbb{C}^{n\times n}$8 is the first column of $A \in \mathbb{C}^{n\times n}$9. For the skew-circulant part, if

$A_{j,k}=a_{j-k}$0

then

$A_{j,k}=a_{j-k}$1

with eigenvalues computable by FFTs. The paper emphasizes that both $A_{j,k}=a_{j-k}$2 and $A_{j,k}=a_{j-k}$3 are computable in $A_{j,k}=a_{j-k}$4 (Liu et al., 2021).

At the operator level, this is a decomposition of linear convolution into periodic and anti-periodic convolution. Toeplitz multiplication $A_{j,k}=a_{j-k}$5 is linear convolution truncated by boundaries. Periodizing the domain introduces wrap-around terms, and the CSCS split partitions them into a symmetric periodic part, carried by $A_{j,k}=a_{j-k}$6, and an antisymmetric anti-periodic part, carried by $A_{j,k}=a_{j-k}$7. In the coefficient formulas these wrap terms appear explicitly as $A_{j,k}=a_{j-k}$8; the identity therefore isolates finite-$A_{j,k}=a_{j-k}$9 boundary effects into two FFT-diagonalizable pieces rather than discarding them.

The same viewpoint clarifies why Fourier methods work so broadly for Toeplitz problems and also why they can fail if the wrong geometry is imposed. A DFT diagonalizes circulants because its basis is periodic; the skew-circulant phase shift implements the corresponding anti-periodic twist. Toeplitz structure is therefore not “approximately Fourier diagonal” in a universal sense: the exact statement is that Toeplitz structure can be recoded into circulant-type pieces whose wrap-around images are explicit and algebraically controlled.

3. Weighted folding, optimal circulants, and asymptotic equivalence

A second major line of work treats aliasing not as an exact decomposition into two pieces, but as a one-piece circulant surrogate. In its simplest form, the first column of the $A = A_c + A_s.$0 circulant induced by a Toeplitz sequence $A = A_c + A_s.$1 is

$A = A_c + A_s.$2

This is the canonical folding of linear-convolution coefficients modulo $A = A_c + A_s.$3. In the frequency domain, the DFT of $A = A_c + A_s.$4 produces circulant eigenvalues that approximate symbol samples, with aliasing error controlled by the tails $A = A_c + A_s.$5. The same mechanism extends to block Toeplitz with Toeplitz blocks, where the BCCB embedding uses the two-dimensional wrap-around

$A = A_c + A_s.$6

(Dykes et al., 2016).

Several standard circulant constructions are weighted variants of this folding. Strang’s preconditioner keeps the nearest Toeplitz diagonals and performs a single wrap. Chan’s Frobenius-optimal circulant uses

$A = A_c + A_s.$7

For real symmetric Toeplitz matrices, the nearest circulant in the Frobenius norm is likewise symmetric, with

$A = A_c + A_s.$8

Its eigenvalues satisfy

$A = A_c + A_s.$9

so the spectrum is the Fejér-kernel smoothing of the Toeplitz symbol sampled on the $c=(a_0,a_1,\dots,a_{n-1})^T$0-point grid. The same Fejér-averaged aliasing formula appears in the optimal circulant projection used for preconditioning functions of Toeplitz matrices (Salahub, 2022, Hon, 2018).

Asymptotic equivalence results show how far these surrogates can be pushed. For uniformly absolutely bounded Hermitian Toeplitz sequences $c=(a_0,a_1,\dots,a_{n-1})^T$1 and derived circulants $c=(a_0,a_1,\dots,a_{n-1})^T$2, square summability of the Toeplitz coefficients implies

$c=(a_0,a_1,\dots,a_{n-1})^T$3

hence $c=(a_0,a_1,\dots,a_{n-1})^T$4. This yields equal distribution of eigenvalues in the collective sense. Individual eigenvalue convergence is stronger: it holds for the three standard derived circulants when $c=(a_0,a_1,\dots,a_{n-1})^T$5, and for the Cesàro circulant under weaker assumptions when the symbol is bounded, Riemann integrable, and has connected essential range. For band Toeplitz matrices the individual error is $c=(a_0,a_1,\dots,a_{n-1})^T$6, and the largest and smallest eigenvalues converge to the essential supremum and infimum of the symbol (Zhu et al., 2016).

A frequent misunderstanding is to treat these statements as interchangeable. They are not. Exact finite-$c=(a_0,a_1,\dots,a_{n-1})^T$7 identities, Frobenius-optimal projections, and asymptotic equivalence belong to different levels of the theory; the existence of a small Frobenius error does not by itself imply individual eigenvalue convergence.

4. Numerical linear algebra: Sylvester solvers, preconditioners, and multigrid

The CSCS iterative method for the Sylvester equation

$c=(a_0,a_1,\dots,a_{n-1})^T$8

assumes $c=(a_0,a_1,\dots,a_{n-1})^T$9 and $r=(a_0,a_{-1},\dots,a_{-(n-1)})$0 are Toeplitz and splits them as

$r=(a_0,a_{-1},\dots,a_{-(n-1)})$1

With positive shift parameters $r=(a_0,a_{-1},\dots,a_{-(n-1)})$2, the method performs two substeps per iteration:

$r=(a_0,a_{-1},\dots,a_{-(n-1)})$3

followed by

$r=(a_0,a_{-1},\dots,a_{-(n-1)})$4

Because $r=(a_0,a_{-1},\dots,a_{-(n-1)})$5 are diagonalizable by FFTs or phase-shifted FFTs, each Sylvester subproblem reduces to diagonal linear systems. The iteration matrix for the Kronecker-sum formulation is

$r=(a_0,a_{-1},\dots,a_{-(n-1)})$6

with spectral radius bounded by

$r=(a_0,a_{-1},\dots,a_{-(n-1)})$7

If one of $r=(a_0,a_{-1},\dots,a_{-(n-1)})$8 is positive definite and the remaining three are positive semidefinite, the CSCS iteration converges. Theorem 4.3 also gives a closed-form parameter choice $r=(a_0,a_{-1},\dots,a_{-(n-1)})$9 minimizing the bound, with the natural practical choice $$\mathrm{CAc} = \frac{c + [0,\operatorname{reverse}(r(2\!:\!n))]}{2}, \qquad \mathrm{CAr} = [r(1)/2,\operatorname{reverse}(\mathrm{CAc}(2\!:\!n))],$$0 when $$\mathrm{CAc} = \frac{c + [0,\operatorname{reverse}(r(2\!:\!n))]}{2}, \qquad \mathrm{CAr} = [r(1)/2,\operatorname{reverse}(\mathrm{CAc}(2\!:\!n))],$$1 and $$\mathrm{CAc} = \frac{c + [0,\operatorname{reverse}(r(2\!:\!n))]}{2}, \qquad \mathrm{CAr} = [r(1)/2,\operatorname{reverse}(\mathrm{CAc}(2\!:\!n))],$$2 have comparable norms. The implementation uses spectral residual computation to avoid explicit dense multiplications, giving $$\mathrm{CAc} = \frac{c + [0,\operatorname{reverse}(r(2\!:\!n))]}{2}, \qquad \mathrm{CAr} = [r(1)/2,\operatorname{reverse}(\mathrm{CAc}(2\!:\!n))],$$3 complexity per FFT per column (Liu et al., 2021).

In ill-posed Toeplitz and BTTB systems, aliasing is used more selectively. Dykes, Noschese, and Reichel build the Frobenius-optimal Chan circulant and then replace the $$\mathrm{CAc} = \frac{c + [0,\operatorname{reverse}(r(2\!:\!n))]}{2}, \qquad \mathrm{CAr} = [r(1)/2,\operatorname{reverse}(\mathrm{CAc}(2\!:\!n))],$$4 smallest-in-magnitude eigenvalues by $$\mathrm{CAc} = \frac{c + [0,\operatorname{reverse}(r(2\!:\!n))]}{2}, \qquad \mathrm{CAr} = [r(1)/2,\operatorname{reverse}(\mathrm{CAc}(2\!:\!n))],$$5,

$$\mathrm{CAc} = \frac{c + [0,\operatorname{reverse}(r(2\!:\!n))]}{2}, \qquad \mathrm{CAr} = [r(1)/2,\operatorname{reverse}(\mathrm{CAc}(2\!:\!n))],$$6

so that the invariant subspace associated with the smallest eigenvalues is almost preserved rather than aggressively preconditioned. Their perturbation analysis leads to the criterion

$$\mathrm{CAc} = \frac{c + [0,\operatorname{reverse}(r(2\!:\!n))]}{2}, \qquad \mathrm{CAr} = [r(1)/2,\operatorname{reverse}(\mathrm{CAc}(2\!:\!n))],$$7

and then sets $$\mathrm{CAc} = \frac{c + [0,\operatorname{reverse}(r(2\!:\!n))]}{2}, \qquad \mathrm{CAr} = [r(1)/2,\operatorname{reverse}(\mathrm{CAc}(2\!:\!n))],$$8. The same paper uses the filtered initial approximation

$$\mathrm{CAc} = \frac{c + [0,\operatorname{reverse}(r(2\!:\!n))]}{2}, \qquad \mathrm{CAr} = [r(1)/2,\operatorname{reverse}(\mathrm{CAc}(2\!:\!n))],$$9

and stops the Krylov process by the discrepancy principle. On the reported image restoration and 1D “gravity” tests, the circulant or BCCB preconditioner roughly halves the iteration count while keeping restoration quality comparable to unpreconditioned runs (Dykes et al., 2016).

In multigrid, aliasing governs coarse-grid symbol formation. For a circulant $$\mathrm{SAc} = \frac{c - [0,\operatorname{reverse}(r(2\!:\!n))]}{2}, \qquad \mathrm{SAr} = [r(1)/2,-\operatorname{reverse}(\mathrm{SAc}(2\!:\!n))].$$0, a projector $$\mathrm{SAc} = \frac{c - [0,\operatorname{reverse}(r(2\!:\!n))]}{2}, \qquad \mathrm{SAr} = [r(1)/2,-\operatorname{reverse}(\mathrm{SAc}(2\!:\!n))].$$1 yields a coarse operator whose symbol is

$$\mathrm{SAc} = \frac{c - [0,\operatorname{reverse}(r(2\!:\!n))]}{2}, \qquad \mathrm{SAr} = [r(1)/2,-\operatorname{reverse}(\mathrm{SAc}(2\!:\!n))].$$2

Equivalently, if $$\mathrm{SAc} = \frac{c - [0,\operatorname{reverse}(r(2\!:\!n))]}{2}, \qquad \mathrm{SAr} = [r(1)/2,-\operatorname{reverse}(\mathrm{SAc}(2\!:\!n))].$$3, then $$\mathrm{SAc} = \frac{c - [0,\operatorname{reverse}(r(2\!:\!n))]}{2}, \qquad \mathrm{SAr} = [r(1)/2,-\operatorname{reverse}(\mathrm{SAc}(2\!:\!n))].$$4. The $$\mathrm{SAc} = \frac{c - [0,\operatorname{reverse}(r(2\!:\!n))]}{2}, \qquad \mathrm{SAr} = [r(1)/2,-\operatorname{reverse}(\mathrm{SAc}(2\!:\!n))].$$5 case suffers from “mirror point” pathology when zeros of $$\mathrm{SAc} = \frac{c - [0,\operatorname{reverse}(r(2\!:\!n))]}{2}, \qquad \mathrm{SAr} = [r(1)/2,-\operatorname{reverse}(\mathrm{SAc}(2\!:\!n))].$$6 occur at points offset by $$\mathrm{SAc} = \frac{c - [0,\operatorname{reverse}(r(2\!:\!n))]}{2}, \qquad \mathrm{SAr} = [r(1)/2,-\operatorname{reverse}(\mathrm{SAc}(2\!:\!n))].$$7; the paper shows that choosing $$\mathrm{SAc} = \frac{c - [0,\operatorname{reverse}(r(2\!:\!n))]}{2}, \qquad \mathrm{SAr} = [r(1)/2,-\operatorname{reverse}(\mathrm{SAc}(2\!:\!n))].$$8 and designing $$\mathrm{SAc} = \frac{c - [0,\operatorname{reverse}(r(2\!:\!n))]}{2}, \qquad \mathrm{SAr} = [r(1)/2,-\operatorname{reverse}(\mathrm{SAc}(2\!:\!n))].$$9 to vanish on the mirror set removes this obstruction. With $$A_c=\operatorname{toeplitz}(\mathrm{CAc},\mathrm{CAr})$$0, the resulting multigrid cycle is optimal in the paper’s cost model (Donatelli et al., 2010). This suggests that Toeplitz-to-circulant aliasing is not merely a device for preconditioning; it is also the algebraic mechanism by which coarse-grid correction is defined.

5. Boundary-aware operator compilation and matrix functions

A recent quantum formulation makes the boundary issue explicit. For the semi-discrete fractional Laplacian on a bounded interval with open, zero-extension boundary conditions, the physical operator $$A_c=\operatorname{toeplitz}(\mathrm{CAc},\mathrm{CAr})$$1 is a Toeplitz truncation of an infinite-lattice convolution kernel $$A_c=\operatorname{toeplitz}(\mathrm{CAc},\mathrm{CAr})$$2. An $$A_c=\operatorname{toeplitz}(\mathrm{CAc},\mathrm{CAr})$$3-point QFT, however, diagonalizes the circulant operator

$$A_c=\operatorname{toeplitz}(\mathrm{CAc},\mathrm{CAr})$$4

not the Toeplitz truncation. The exact aliasing proposition is

$$A_c=\operatorname{toeplitz}(\mathrm{CAc},\mathrm{CAr})$$5

so

$$A_c=\operatorname{toeplitz}(\mathrm{CAc},\mathrm{CAr})$$6

Zero-padding to size $$A_c=\operatorname{toeplitz}(\mathrm{CAc},\mathrm{CAr})$$7 and compressing back gives

$$A_c=\operatorname{toeplitz}(\mathrm{CAc},\mathrm{CAr})$$8

with

$$A_c=\operatorname{toeplitz}(\mathrm{CAc},\mathrm{CAr})$$9

by the Schur test. For the classical second-difference Laplacian, the kernel is finitely supported and the aliasing reduces to corner wraps only; padding to any $a_{i-j+\ell N}$00 eliminates them exactly on the compressed subspace. One call to the resulting block-encoding has cost

$a_{i-j+\ell N}$01

(Javanmard et al., 16 May 2026).

Aliasing also underlies preconditioners for functions of Toeplitz matrices. For a Hermitian Toeplitz matrix $a_{i-j+\ell N}$02 generated by a positive Wiener-class symbol $a_{i-j+\ell N}$03, the optimal circulant $a_{i-j+\ell N}$04 has eigenvalues given by Fejér-aliased samples of $a_{i-j+\ell N}$05, while the superoptimal circulant $a_{i-j+\ell N}$06 has eigenvalues

$a_{i-j+\ell N}$07

that is, gridwise ratios of aliased samples of $a_{i-j+\ell N}$08 and $a_{i-j+\ell N}$09. For analytic $a_{i-j+\ell N}$10, the paper proves low-rank-plus-small-norm decompositions of $a_{i-j+\ell N}$11 and shows that the absolute value superoptimal preconditioner $a_{i-j+\ell N}$12 yields spectra of $a_{i-j+\ell N}$13 clustered around $a_{i-j+\ell N}$14. The same construction extends to BTTB matrices with optimal BCCB preconditioners, although the multilevel case has $a_{i-j+\ell N}$15 outliers rather than a fixed number (Hon, 2018).

6. Parity splittings, random matrices, and conceptual scope

A particularly transparent structured example is Driessel’s near-Toeplitz tridiagonal matrix

$a_{i-j+\ell N}$16

where $a_{i-j+\ell N}$17 is the basic circulant, $a_{i-j+\ell N}$18 the basic skew-circulant, and $a_{i-j+\ell N}$19 are the projections onto even and odd subspaces with respect to the exchange matrix $a_{i-j+\ell N}$20. On $a_{i-j+\ell N}$21, $a_{i-j+\ell N}$22 acts exactly as the periodic first-difference operator $a_{i-j+\ell N}$23; on $a_{i-j+\ell N}$24, it acts as the anti-periodic first-difference operator $a_{i-j+\ell N}$25. The discrepancy between Toeplitz ends and wrap-around corners vanishes on the corresponding parity subspace (Driessel, 2011).

The same circulant/skew-circulant algebra has recently been used in random matrix theory. For non-symmetric Toeplitz matrices, the identity

$a_{i-j+\ell N}$26

expresses Toeplitz structure as a sum of a circulant and a skew-circulant. For Hankel matrices,

$a_{i-j+\ell N}$27

with $a_{i-j+\ell N}$28 reverse circulant and $a_{i-j+\ell N}$29 left skew-circulant. Since the circulant-type components have tractable $a_{i-j+\ell N}$30-limits, the paper derives $a_{i-j+\ell N}$31-convergence results for random Toeplitz and Hankel matrices from these deterministic identities. In particular, symmetric Toeplitz matrices converge in $a_{i-j+\ell N}$32-distribution to the sum of two self-adjoint Gaussian variables, while random Hankel matrices converge to the sum of two self-adjoint symmetrized Rayleigh variables (Bose et al., 15 May 2026).

Three distinctions delimit the scope of the subject. First, exact finite-$a_{i-j+\ell N}$33 decompositions such as $a_{i-j+\ell N}$34 or $a_{i-j+\ell N}$35 are not the same as optimal one-piece circulant approximations or asymptotic equivalence theorems (Zhu et al., 2016). Second, a finite DFT or QFT diagonalizes a circulant surrogate by construction; it does not diagonalize an open-boundary Toeplitz truncation unless an additional boundary adapter, such as zero-padding and compression, is introduced (Javanmard et al., 16 May 2026). Third, in inverse problems the smallest-eigenvalue subspace is often the noise-sensitive part of the model, so an effective circulant preconditioner may need to preserve that subspace rather than flatten it (Dykes et al., 2016).

Taken together, these results define the Toeplitz-to-Circulant Aliasing Identity as a unifying operator principle: linear-convolution structure can be folded into periodic or anti-periodic convolution, either exactly or asymptotically, provided the wrap-around images are represented explicitly. The resulting diagonalizability by FFT, phase-shifted FFT, or QFT is not an incidental computational convenience; it is the direct algebraic consequence of how Toeplitz boundary data are aliased into circulant geometry.