archimedes.measure.HermiteMeasure

class archimedes.measure.HermiteMeasure

Measure for the physicists’ Hermite polynomial family.

Weight \(w(x) = e^{-x^2}\) on \((-\infty, \infty)\).

The associated orthogonal polynomials are the physicists’ Hermite polynomials \(H_n(x)\) (as opposed to the probabilists’ convention used by HermiteNormMeasure, which instead uses weight \(e^{-x^2/2}\)). The zeroth moment of the weight is \(\int_{-\infty}^\infty e^{-x^2} \, dx = \sqrt{\pi}\).

__init__()

Methods

__init__()
affine_params([mean, std])	Map the reference weight onto a location-scaled Gaussian-shaped weight \(w(x) = \exp(-((x - \mathrm{mean})/ \mathrm{std})^2)\).
mass(*args, **kwargs)	Total mass of the measure mapped by affine_params(*args, **kwargs).
weight(x)	Reference weight function \(w(x) = e^{-x^2}\), evaluated at x.

Attributes

reference_mass	\(\int_{-\infty}^\infty e^{-t^2} \, dt = \sqrt{\pi}\).
support	Support \((-\infty, \infty)\).
uniform_weight	True if the weight is constant (weight(x) == weight(y)) for every x, y in support -- e.g. true for Legendre, false for Jacobi (singular at the endpoints) or Hermite/Laguerre (unbounded support).