The multi-dimensional transforms of FFTW, in general, compute simply the separable product of the given 1d transform along each dimension of the array. Since each of these transforms is unnormalized, computing the forward followed by the backward/inverse multi-dimensional transform will result in the original array scaled by the product of the normalization factors for each dimension (e.g. the product of the dimension sizes, for a multi-dimensional DFT).

The definition of FFTW's multi-dimensional DFT of real data (r2c)
deserves special attention. In this case, we logically compute the full
multi-dimensional DFT of the input data; since the input data are purely
real, the output data have the Hermitian symmetry and therefore only one
non-redundant half need be stored. More specifically, for an n_{0} × n_{1} × n_{2} × … × n_{d-1} multi-dimensional real-input DFT, the full (logical) complex output array
*Y*[*k*_{0}, *k*_{1}, ...,
*k*_{d-1}]has the symmetry:
*Y*[*k*_{0}, *k*_{1}, ...,
*k*_{d-1}] = *Y*[*n*_{0} -
*k*_{0}, *n*_{1} - *k*_{1}, ...,
*n*_{d-1} - *k*_{d-1}]^{*}(where each dimension is periodic). Because of this symmetry, we only
store the
*k*_{d-1} = 0...*n*_{d-1}/2+1elements of the *last* dimension (division by 2 is rounded
down). (We could instead have cut any other dimension in half, but the
last dimension proved computationally convenient.) This results in the
peculiar array format described in more detail by Real-data DFT Array Format.

The multi-dimensional c2r transform is simply the unnormalized inverse of the r2c transform. i.e. it is the same as FFTW's complex backward multi-dimensional DFT, operating on a Hermitian input array in the peculiar format mentioned above and outputting a real array (since the DFT output is purely real).

We should remind the user that the separable product of 1d transforms
along each dimension, as computed by FFTW, is not always the same thing
as the usual multi-dimensional transform. A multi-dimensional
`R2HC`

(or `HC2R`

) transform is not identical to the
multi-dimensional DFT, requiring some post-processing to combine the
requisite real and imaginary parts, as was described in The Halfcomplex-format DFT. Likewise, FFTW's multidimensional
`FFTW_DHT`

r2r transform is not the same thing as the logical
multi-dimensional discrete Hartley transform defined in the literature,
as discussed in The Discrete Hartley Transform.