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This paper introduces a new method for adaptively building a multivariate C° spline approximation from scattered samples of an unknown function. The central feature of the method is a means for adaptively tesselating an approximation space to form a multidimensional mesh over which the spline fitting then occurs. The mesh used is a Delaunay tesselation of the approximation space whose vertices lie at a subset of the scattered sample locations. The specific subset of sample locations used is adaptively determined by repeated overfitting and simplification of the resulting spline approximation. Overfitting and simplification is an attractive paradigm for high-dimensional approximation problems because it provides a means for forming an approximation that is complex only in regions where the scattered sample data provide sufficient evidence of complexity in the underlying unknown function. Overfitting and simplification is effectively exploited in this new approach as the function representation used is not subject to certain recursive dependencies. The properties of the new technique are demonstrated in the context of an easily visualized bivariate classification problem, The technique is then applied to a 10-dimensional clinical ECG classification problem, and the results are compared to those obtained with a perceptron based neural network.