  $R$-limited functions are multivariate generalization of band-limited
functions whose Fourier transforms are supported within a compact region
$R\subset\mathbb{R}^{n}$. In this work, we generalize sampling and
interpolation theorems for band-limited functions to $R$-limited functions.
More precisely, we investigated the following question: "For a function
compactly supported within a region similar to $R$, does there exist an
$R$-limited function that agrees with the function over its support for a
desired accuracy?". Starting with the Fourier domain definition of an
$R$-limited function, we write the equivalent convolution and a discrete
Fourier transform representations for $R$-limited functions through
approximation of the convolution kernel using a discrete subset of Fourier
basis. The accuracy of the approximation of the convolution kernel determines
the accuracy of the discrete Fourier representation. Construction of the
discretization can be achieved using the tools from approximation theory as
demonstrated in the appendices. The main contribution of this work is proving
the equivalence between the discretization of the Fourier and convolution
representations of $R$-limited functions. Here discrete convolution
representation is restricted to shifts over a compactly supported region
similar to $R$. We show that discrete shifts for the convolution representation
are equivalent to the spectral parameters used in discretization of the Fourier
representation of the convolution kernel. This result is a generalization of
the cardinal theorem of interpolation of band-limited functions. The error
corresponding to discrete convolution representation is also bounded by the
approximation of the convolution kernel using discretized Fourier basis.
