Chebyshev1st

Chebyshev1st(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 \[ \begin{align} p_{0}(x) &= 1, \\ p_{k}(x) &= \sqrt{2}\cos(k\arccos(x)), \qquad k = 1, 2, \dots, n. \end{align} \] The polynomials are orthogonal with respect to the (normalised) weighting function given by \[ \lambda(x) = \frac{1}{\pi\sqrt{1-x^{2}}}. \]

References

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

Cui, T, Dolgov, S and Zahm, O (2023). Self-reinforced polynomial approximation methods for concentrated probability densities. arXiv preprint.