Theory

PETAL2D is built around a small set of exact Fourier facts plus explicit numerical approximations. The goal of this section is to make every reported quantity mathematically transparent.

• Polar harmonic expansion
  • Function space and convention
  • Physical meaning of \(m\)
  • Mode power
  • Why retain radial profiles?
  • Point-group fingerprints
• Discrete angular sampling and Nyquist structure
  • Representable FFT indices
  • Aliasing
  • Real fields
  • Choosing Ntheta
• Radial cutoff diagnostics
  • Power-support radius
  • Amplitude-support radius
  • Final cutoff radius
  • Why two criteria are necessary
• Proofs
  • Polar Fourier expansion and Parseval identity
  • Exact reconstruction error
  • Optimal power-ranked truncation
  • Real-valued fields and conjugate pairs
  • Rotation covariance
  • Rotational selection rules
  • Discrete angular aliasing
  • Angular-momentum interpretation for a wavefunction
  • Translation of the power centroid

The Proofs page gives formal proofs. The other pages emphasize physical intuition and numerical interpretation.