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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.