archimedes.measure.HermiteNormMeasure

class archimedes.measure.HermiteNormMeasure

Measure for the probabilists’ Hermite polynomial family.

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

The associated orthogonal polynomials are the probabilists’ Hermite polynomials \(\mathit{He}_n(x)\) (as opposed to the physicists’ convention used by HermiteMeasure, with weight \(e^{-x^2}\)). Up to normalization, this weight is exactly the density of a standard normal distribution: \(e^{-x^2/2} = \sqrt{2\pi} \, \phi(x)\), where \(\phi\) is the standard normal PDF. The zeroth moment of the weight is \(\int_{-\infty}^\infty e^{-x^2/2} \, dx = \sqrt{2\pi}\).

__init__()

Methods

__init__()
affine_params([mean, std])	Map the reference weight onto a Gaussian weight with the given mean and standard deviation: \(w(x) = \exp(-(x - \mathrm{mean})^2 / (2 \, \mathrm{std}^2))\).
mass(*args, **kwargs)	Total mass of the measure mapped by affine_params(*args, **kwargs).
weight(x)	Weight function \(w(x) = e^{-x^2/2}\), evaluated at x.

Attributes

reference_mass	int_{-infty}^infty e^{-t^2/2} , dt = sqrt{2pi}.
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).