Spectral methods and aliasing

Use N equispaced points x_j=2pi j/N on a periodic interval[0,2pi]. For odd N, retain the integer modes from−(N−1)/2 through(N−1)/2, avoiding an even-grid Nyquist convention. The forward transform divides by N and the inverse sum has no extra factor. Differentiating the trigonometric interpolant multiplies each coefficient by ik. Changing transform normalization without adjusting convolution or Parseval factors can introduce an incorrect N into a PDE operator or energy calculation.

The derivative is exact for a trigonometric polynomial whose frequencies lie in that retained set, up to floating-point effects. For a smooth periodic function with infinitely many modes, derivative error depends on coefficient decay, hence on regularity. Analyticity can yield exponential convergence under suitable periodic extension, whereas finite smoothness gives algebraic rates. A jump or a nonperiodic endpoint mismatch destroys the assumptions behind rapid spectral convergence and can produce global oscillatory artifacts even on a very fine grid.

At these grid points, exp(i(k+qN)x_j) equals exp(ikx_j) for every integer q. An unresolved continuum mode is therefore folded into a retained representative. For functions with an absolutely summable Fourier series, the sampled DFT coefficient is the sum of continuum coefficients in that congruence class. This identity explains why a derivative computed after sampling can disagree dramatically with the sampled exact derivative: multiplication by the retained k is not multiplication by the original k+qN.

Aliasing can also be created from initially resolved data. Multiplying two trigonometric polynomials adds their frequencies. If the product is evaluated on the original grid, new modes outside the representable band fold back into low modes. Thus a nonlinear PDE can generate incorrect retained coefficients even when every factor was initially resolved. Projecting an accurately resolved product and aliasing an underresolved product are different operations; one cannot repair the second simply by deleting a few high-frequency coefficients afterward.

Zero-padding known Fourier coefficients to a sufficiently fine grid allows a nonlinear product to be evaluated with more frequency room before projection back to the retained band. For quadratic nonlinearities, the familiar three-halves padding strategy uses a consistent retained-mode and endpoint convention. Its purpose is to prevent product frequencies from wrapping into the coefficients one keeps, not to infer high-frequency initial data that were never measured. Higher-degree nonlinearities or different retained bands require their own padding calculation.

For example, on nine points, squaring the resolved cosine 3x creates a cosine 6x term that aliases back to cosine 3x. Evaluating that known polynomial on fifteen points resolves its product frequencies through six, after which projection to the original modes removes the six-frequency term instead of folding it. This specific calculation is an exact polynomial statement. It does not promise that a general nonlinear solution remains smooth, that a time integrator is stable, or that filtering preserves every conservation property of the PDE.