Networks of wireless micro-sensors are envisioned to be the prominent choice for on-site monitoring of physical locations. A wide range of practical applications has been conceived and studied in recent years for this engineering regime: habitat and wildlife monitoring, smart buildings and disaster response are only a few representative examples. However, there are also unique challenges faced by the sensor network paradigm: energy resources for individual sensors are limited. Efficient approaches are necessary to ensure prolonged autonomous operation of the network, while still providing quality of service to the user application at all times.;Here, we focus on situations where the wireless sensor network functions as a distributed sampling system and sensors periodically sample a physical phenomenon of interest, e.g. temperature. Samples are then used to construct a spatially continuous estimate of the phenomenon through interpolation, over time.;We examine two distinct classes of practical sampling-interpolation scenarios. In the first one we are given a large ensemble of sensors which have already been deployed. The goal is then to reactively devise a maximum number of disjoint subsets of sensors, such that data from each of them can individually support the desired interpolation accuracy. Energy efficiency is achieved by reducing the amount of data packets communicated across the network. In the second one we have to proactively manage deployment of the network from scratch. The objective is then to use a minimum number of sensors so as to again support the desired interpolation accuracy. Cost effectiveness is achieved here by using a smaller network to begin with.;To tackle the challenges of these scenarios we utilize the Hilbert space of second order random variables and define interpolation quality on the basis of Mean Squared Error (MSE). Times series of values measured at individual sensors can provide finite dimensional approximations of these random variables and facilitate algebraic manipulations within the Hilbert space framework. The associated covariance matrix succinctly captures sensor correlations and enables novel solutions to the aforementioned problems. Through extensive simulations on synthetic and real sensor network data our proposed solutions are shown to possess strong advantages compared to other approaches.
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