# fourierbasis

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##### Fourier Basis Functions

Evaluates the Fourier basis functions on a $$d$$-dimensional box with $$d$$-dimensional frequencies $$k_i$$ at the $$d$$-dimensional coordinates $$x_j$$.

##### Usage
fourierbasis(x, k, win = boxx(rep(list(0:1), ncol(k))))
##### Arguments
x

Coordinates. A data.frame or matrix with $$m$$ rows and $$d$$ columns giving the $$d$$-dimensional coordinates.

k

Frequencies. A data.frame or matrix with $$n$$ rows and $$d$$ columns giving the frequencies of the Fourier-functions.

win

window (of class "owin", "box3" or "boxx") giving the $$d$$-dimensional box domain of the Fourier functions.

##### Details

The result is an $$n$$ by $$m$$ matrix where the $$(i,j)$$'th entry is the $$d$$-dimensional Fourier basis function with frequency $$k_i$$ evaluated at the point $$x_j$$, i.e., $$\frac{1}{|W|} \exp(2\pi i <k_i,x_j>/|W|)$$ where $$<\cdot,\cdot>$$ is the $$d$$-dimensional inner product and $$|W|$$ is the volume of the domain (window/box). Note that the algorithm does not check whether the coordinates given in x are contained in the given box. Actually the box is only used to determine the volume of the domain for normalization.

##### Value

An n by m matrix of complex values.

• fourierbasis
##### Examples
# NOT RUN {
## 27 rows of three dimensional Fourier frequencies:
k <- expand.grid(-1:1,-1:1, -1:1)
## Two random points in the three dimensional unit box:
x <- rbind(runif(3),runif(3))
## 27 by 2 resulting matrix:
v <- fourierbasis(x, k)