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A quadratic Fourier product aliases two high modes on an even grid

Example

On the $N = 4$ periodic grid, let $p (x) = \cos x + \cos (2 x)$ . The continuous square has modes through frequency four, but its grid interpolant is

I_{4} (p^{2}) = \frac{3}{2} + 2 \cos x + \frac{1}{2} \cos (2 x) .

The frequency-three mode aliases to frequency one, and the frequency-four mode aliases to the constant mode. This exact sample identity is not a de-aliasing or nonlinear stability result.

Facts & Assumptions

Given: The four-point grid $x_{j} = π j / 2$ , the interpolant convention, and $p (x) = \cos x + \cos (2 x)$ .

[F1]

Periodic Fourier collocation grid and even-grid Nyquist convention: for $N = 4$ , the interpolation space is spanned over $R$ by $1, \cos x, \sin x, \cos (2 x)$ .

[F2]

Periodic Fourier modes alias modulo the grid size: modes whose integer frequencies are congruent modulo $4$ have the same samples, and sampled products have frequency equal to the sum modulo $4$ .

[F3]

The addition formulas for sine and cosine: $\cos u \cos v = \frac{1}{2} (\cos (u + v) + \cos (u - v))$ .

[F4]

Quarter-turn values and shifts by pi/2 and pi: on $x_{j} = π j / 2$ , $\cos (4 x_{j}) = 1$ and $\cos (3 x_{j}) = \cos (x_{j})$ .

Proof

1.1F3algebra

By [F3], $p (x)^{2} = \cos^{2} x + 2 \cos x \cos (2 x) + \cos^{2} (2 x) = 1 + \cos x + \frac{1}{2} \cos (2 x) + \cos (3 x) + \frac{1}{2} \cos (4 x) .$

2.1givenF1F4step 1.1algebra

At every grid node [F4] turns this into $p (x_{j})^{2} = \frac{3}{2} + 2 \cos (x_{j}) + \frac{1}{2} \cos (2 x_{j})$ . The right side belongs to [F1]'s interpolation space.

3.1F1F2step 1.1step 2.1algebra∎

The modes at frequencies $3$ and $4$ in step 1.1 are identified on the grid by the congruences $3 \equiv - 1 (m o d 4)$ and $4 \equiv 0 (m o d 4)$ , exactly as [F2] states. Thus the displayed interpolant has the same samples as $p^{2}$ , with the two alias contributions combined.

Depends on

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Sources

Benjamin Seibold, MIT 18.336 Lecture 24: Pseudospectral Methods (standard reference, not scraped)
Lloyd N. Trefethen, Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations, Ch. 7 (standard reference, not scraped)