FourierBasis {USP}

Fourier basis functions

Description

Computes the values of the one-dimensional Fourier basis functions at a vector of locations x and with a vector of frequencies m. The scaling factor of 2\pi is included, so that the function returns, e.g., \sqrt{2} \cos(2\pi m x).

Usage

FourierBasis(a, m, x)

Arguments

a

Sine or cosine; a=0 gives cosine and a=1 gives sine.

m

Vector of frequencies m.

x

Vector of locations x.

Value

Returns the values of \sqrt{2} \cos(2\pi m x).

References

Berrett TB, Kontoyiannis I, Samworth RJ (2021). “Optimal rates for independence testing via U-statistic permutation tests.” Annals of Statistics, to appear.

Examples

e=FourierBasis(1,1:100,0.01); plot(0.01*(1:100),e,type="l")
e=FourierBasis(0,1,0.01*(1:100)); plot(0.01*(1:100),e,type="l")
FourierBasis(1,1:3,0.1*(1:10))