  Conventional sampling and interpolation commonly rely on discrete
measurements. In this paper, we develop a theoretical framework for
extrapolation of signals in higher dimensions from knowledge of the continuous
waveform on bounded high-dimensional regions. In particular, we propose an
iterative method to reconstruct bandlimited multidimensional signals based on
truncated versions of the original signal to bounded regions---herein referred
to as continuous measurements. In the proposed method, the reconstruction is
performed by iterating on a convex combination of region-limiting and
bandlimiting operations. We show that this iteration consists of a firmly
nonexpansive operator and prove strong convergence for multidimensional
bandlimited signals. In order to improve numerical stability, we introduce a
regularized iteration and show its connection to Tikhonov regularization. The
method is illustrated numerically for two-dimensional signals.
