R: Fourier Basis Functions

fourierbasis {spatstat} | R Documentation

## 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))))
      fourierbasisraw(x, k, boxlengths)

### Arguments

`x` | Coordinates. A `data.frame` or matrix with n rows and d columns giving the d-dimensional coordinates.
`k` | Frequencies. A `data.frame` or matrix with m 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.
`boxlengths` | numeric giving the side lengths of the box domain of the Fourier functions.

### Details

The result is an m by n 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.,

1/sqrt(|W|) * exp(2*pi*i*(k_{i,1}*x_{j,1}/L_1 + ... + k_{i,d}*x_{j,d}/L_d))

where L_l, l=1,...,d are the box side lengths 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 side lengths and volume of the domain for normalization.

The stripped down faster version `fourierbasisraw` doesn't do checking or conversion of arguments and requires `x` and `k` to be matrices.

### Value

An `m` by `n` matrix of complex values.

### Author(s)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au

Rolf Turner r.turner@auckland.ac.nz

and Ege Rubak rubak@math.aau.dk

### 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)

[Package spatstat version 1.63-3 Index]