Optimal Rotational Smoothing on S1: Why Only the Poisson Kernel Survives on the Circle

Circular signals—angles, phases, orientations—pervade modern science and engineering yet resist straightforward linear smoothing techniques. Context. Circular data arise in disciplines ranging from wind forecasting to phase-unwrapping and cryo-EM. Problem. Despite a century of practice, no consensus exists on which smoothing kernel on the unit circle achieves the optimal balance between symmetry, stability and spectral localisation. Method. We formulate six axioms—(1) reality & evenness, (2) unit mass, (3) an analytic strip with simple poles, (4) a single inflection, (5) positive-definiteness, and (6) a half-height bandwidth criterion—and fuse contour integration, Paley–Wiener theory and Bochner–Herglotz positivity to analyse their joint implications. A residue calculation converts the analytic-strip constraint into an exponential Fourier envelope, while positivity confines residue phases, and normalisation plus curvature conditions fix the remaining scalar. Result. The only kernel satisfying all six axioms is the Poisson (order-1 Butterworth) family Kₐ(φ) = sinh a ⁄ (cosh a − cos φ), a > 0, uniquely determined up to its bandwidth parameter. Implications. Numerical experiments in denoising, spectral-leakage suppression and wind-direction forecasting confirm that once the axioms are accepted, practitioners should “pick their a, not their window,” and adopt Kₐ as the default circular smoother.