Legendre
self, order: int) Legendre(
Legendre polynomials.
Parameters
order : int
-
The maximum order of the polynomials, \(n\).
Notes
The Legendre polynomials, defined on \((-1, 1)\), are given by the recurrence relation (Boyd 2001) \[ (k+1)\hat{p}_{k+1}(x) = (2k+1)x\hat{p}_{k}(x) - k\hat{p}_{k-1}(x), \qquad k = 1, 2, \dots, n-1, \] where \(\hat{p}_{0}(x) = 1, \hat{p}_{1}(x) = x\). The corresponding normalised polynomials are given by \[ p_{k}(x) := \frac{\hat{p}_{k}(x)}{2k+1}, \qquad k = 0, 1, \dots, n. \]
The polynomials are orthonormal with respect to the (normalised) weighting function given by \[ \lambda(x) = \frac{1}{2}. \]
We use Chebyshev polynomials of the second kind to represent the (conditional) CDFs corresponding to the Legendre representation of (the square root of) the target density function.