Legendre

Legendre(order: int)

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 \[ (k+1)\hat{p}_{k+1}(x) = (2k+1)x\hat{p}_{k}(x) - k\hat{p}_{k-1}(x), \qquad k = 1, 2, \dots, n-1, \] 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.

References

Boyd, JP (2001, Appendix A.2). Chebyshev and Fourier spectral methods. Lecture Notes in Engineering, Volume 49.