NDFT

Let the torus

$${\mathbb{T}}^d:= \left\{ \mbox{\boldmath {${x}$}}=\left(x_t\right... ...in \mathbb{R}^d: - \frac{1}{2} \le x_t < \frac{1}{2},\; t=0,\hdots,d-1 \right\}$$

of dimension $d$ be given. It will serve as domain where the non equispaced knots ${x}$ are taken from. Thus, the sampling set is given by ${\cal X}:=\{$${x}$$_j \in {\mathbb{T}}^d:\,j=0,\hdots,M-1\}$.

The space of all ($d$-variate, one-periodic) functions $f: {\mathbb{T}}^d \rightarrow \mathbb{C}$ is restricted to the space of $d$-variate trigonometric polynomials

$${\rm span}\left({\rm e}^{-2\pi{\mbox{\scriptsize {i}}} \mbox{\bol... ...}:\,\mbox{\boldmath {${k}$}} \in I_{\mbox{\boldmath\scriptsize {${N}$}}}\right)$$

with degree $N_t\; (t=0,\hdots,d-1)$ in the $t$-th dimension. Possible frequencies ${k}$ are collected in the multi index set

$$I_{\mbox{\boldmath\scriptsize {${N}$}}} := \left\{ \mbox{\boldmat... ...mathbb{Z}^d: - \frac{N_t}{2} \le k_t < \frac{N_t}{2} ,\;t=0,\hdots,d-1\right\},$$

where ${N}$$=\left(N_t\right)_{t=0,\hdots,d-1}$ is the multi bandlimit. The dimension of the space of $d$-variate trigonometric polynomials is given by $N_{\text{\tiny $\Pi$}}=\prod\limits_{t=0}^{d-1} N_t$.

The inner product between the frequency ${k}$ and the time/spatial knot ${x}$ is defined in the usual way by ${k}$   ${x}$$:=k_0 x_0 + k_1 x_1 +\hdots+ k_{d-1} x_{d-1}$. Furthermore, two vectors may be linked by the pointwise product ${\sigma}$$\odot$   ${N}$$:=\left(\sigma_0 N_0, \sigma_1 N_1, \hdots, \sigma_{d-1} N_{d-1}, \right)^{\rm T}$ with its inverse ${N}$$^{-1}:=\left(\frac{1}{N_0}, \frac{1}{N_1}, \hdots, \frac{1}{N_{d-1}} \right)^{\rm T}$.

For clarity of presentation the multiindex ${k}$ adresses elements of vectors and matrices as well, i.e., the plain index $k_{\text{\tiny $\Pi$}}:=\sum_{t=0}^{d-1} (k_t+\frac{N_t}{2}) \prod_{t'=t+1}^{d-1} N_{t'}$ is not used here.