Fourier

Fourier(self, order: int)

Fourier polynomials.

Parameters

order : int

The number of sine functions the basis is composed of. The total number of basis functions, \(n\), is equal to 2*order+2.

Notes

The Fourier basis for the interval \([-1, 1]\), with cardinality \(n\), is given by (Boyd 2001; Cui and Dolgov 2022) \[ \left\{1, \sqrt{2}\sin(\pi x), \dots, \sqrt{2}\sin(k \pi x), \sqrt{2}\cos(\pi x), \dots, \sqrt{2}\cos(k \pi x), \sqrt{2}\cos(n \pi x / 2)\right\}, \] where \(k = 1, 2, \dots, \tfrac{n}{2}-1\).

The basis functions are orthonormal with respect to the (normalised) weight function given by \[ \lambda(x) = \frac{1}{2}. \]

References

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

Cui, Tiangang, and Sergey Dolgov. 2022. “Deep Composition of Tensor-Trains Using Squared Inverse Rosenblatt Transports.” Foundations of Computational Mathematics 22 (6): 1863–1922. https://doi.org/10.1007/s10208-021-09537-5.