Advantages of using cubic splines to expand functions

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I've recently determined the deuteron's binding energy using cubic B-Splines to expand the system of coupled differential equations I obtained for my problem. This method of expanding functions using such basis was suggested by a professor of mine and I arrived to quite satisfying results.

However I would like to understand better the advantages of these particular splines.

For example, I've read something about the fact these splines have "local minimal support". What exactly is that? Does it relate to the fact that I've defined the splines piecewise?