spatstat (version 1.48-0)

fourierbasis: Fourier Basis Functions

Description

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.

Value

n by m matrix of complex values.

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., $$ \frac{1}{|W|} \exp(2\pi i /|W|) $$ where $*$ 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.

Examples

## 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)
head(v)