Abstract
The properties of best nonlinear approximations with respect to a generalized integral norm on an interval are studied. A necessary condition for an approximation to be locally best is obtained. The interpolatory properties of best approximations are related to the dimension of a Haar subspace in the tangent space. A sufficient condition for an approximation to be best only to itself is given for a class of norms including the L p norms, 1 < p < ∞. A sufficient condition for the set of points at which the given approximated function and an approximant agree to be of positive measure is given. The results are applied to approximation by exponential families V n : in the case of L p approximation, 1 < p < ∞, degenerate approximations are best only to themselves and the error of a best approximation is either identically zero or has 2 n sign changes.
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