Fourier

Fourier(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 \[ \left\{1, \sqrt{2}\sin(\pi x), \dots, \sqrt{2}\sin(k \pi x), \sqrt{2}\cos(\pi x), \dots, \sqrt{2}\cos(k \pi x), \sqrt{2}\cos(n \pi x / 2)\right\}, \] 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, JP (2001, Section 4.5). Chebyshev and Fourier spectral methods. Lecture Notes in Engineering, Volume 49.

Cui, T and Dolgov, S (2022). Deep composition of Tensor-Trains using squared inverse Rosenblatt transports. Foundations of Computational Mathematics 22, 1863–1922.