In a multiple sensor system, the sensor Sj , j=1,2, . . . ,N, outputs Y( j )?R in response to input X?[0,1], according to an unknown probability distribution PY(j )|X . The problem is to estimate a fusion function f:RN?[0,1], based on a training sample, such that the expected square error is minimized over a family of functions F that constitutes a finite-dimensional vector space. The function f* that exactly minimizes the expected error cannot be computed since the underlying distributions are unknown, and only an approximation f to f* is feasible. We estimate the sample size sufficiently to ensure that an estimator f that minimizes the empirical square error provides a close approximation to f* with a high probability. The advantages of vector space methods are twofold: (1) the sample size estimate is a simple function of the dimensionality of F and (2) the estimate f can be easily computed by the well-known least square methods in polynomial time. The results are applicable to the classical potential function method as well as to a recently proposed class of sigmoidal feedforward neural networks.