Chebyshev1st

Chebyshev1st(self, order: int)

Chebyshev polynomials of the first kind.

Parameters

order : int

The maximum order of the polynomials.

Notes

The (normalised) Chebyshev polynomials of the first kind, defined on \((-1, 1)\), are given by (Boyd 2001; Cui, Dolgov, and Zahm 2023) \[ \begin{align} p_{0}(x) &= 1, \\ p_{k}(x) &= \sqrt{2}\cos(k\arccos(x)), \qquad k = 1, 2, \dots, n. \end{align} \] The polynomials are orthogonal with respect to the (normalised) weighting function given by \[ \lambda(x) = \frac{1}{\pi\sqrt{1-x^{2}}}. \]

References

Boyd, John P. 2001. Chebyshev and Fourier Spectral Methods. https://link.springer.com/book/9783540514879.

Cui, Tiangang, Sergey Dolgov, and Olivier Zahm. 2023. “Self-Reinforced Polynomial Approximation Methods for Concentrated Probability Densities.” https://arxiv.org/abs/2303.02554.