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The best
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approximation of the Heaviside function and the best
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approximation of multiscale univariate datasets by a cubic spline have a Gibbs phenomenon near the discontinuity. We show by numerical experiments that the Gibbs phenomenon can be reduced by using
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spline fits which are the best
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approximations in an appropriate spline space obtained by the union of
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interpolation splines. We prove here the existence of
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spline fits for function approximation which has never previously been done to the best of our knowledge. A major disadvantage of this technique is an increased computation time. Thus, we propose a sliding window algorithm on seven nodes which is as efficient as the global method both for functions and datasets with abrupt changes of magnitude, but within a linear complexity on the number of spline nodes.