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On uniqueness of best L 1-approximation in sobolev spaces

In this paper we investigate the uniqueness of best approximation of continuous functions on [-a a] in the norm , by elements of Haar and A-spaces. It turns out that the size of the interval is crucial for uniqueness. In most cases we shall determine the exact set of a's for which uniqueness holds. We also show that in the case when r = 1 the approximation by Haar spaces yields uniqueness for every a.