Decomposition of Solar Wind Velocity Distribution Functions with Orthogonal Polynomials

We present a framework for decomposing solar-wind velocity distribution functions (VDFs) using orthogonal polynomial bases. We aim to establish a practical procedure for applying polynomial decompositions to in-situ spacecraft VDFs and to clarify how the resulting spectra of expansion-coefficient power can be used for noise reduction, VDF reconstruction, and diagnostics of velocity-space structure. The method represents measured VDF structure with Hermite-Hermite and Hermite-Laguerre expansions, providing a nonparametric description of departures from Maxwellians, such as anisotropy, skewness, beams, and suprathermal tails. Expansion coefficients are estimated by Gaussian-weighted quadrature after interpolation of measured distributions onto polynomial nodes. We demonstrate several applications of polynomial decomposition to Solar Orbiter, Parker Solar Probe, and Magnetospheric Multiscale 1 measurements, including noise identification through high-order spectral flattening, noise-reduced VDF reconstruction, and characterization of VDF-structure variations under different plasma conditions, e.g., turbulent solar-wind streams and shocks. For instance, noise-reduced reconstructed VDFs can provide smoother estimates of distinct ion populations and VDF gradients. Examples from solar-wind streams and collisionless-shock crossings further show that the resulting spectra respond to changes in parallel and perpendicular VDF structure, illustrating their potential for comparing kinetic modifications under different plasma conditions. Overall, orthogonal-polynomial decomposition provides a bridge between measured particle distributions and kinetic plasma physics by converting complex VDF morphology into quantitative velocity-space spectra.