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The title says it all! When using orthogonal series expansions like the Gram-Charlier expansion to approximate probability density function, a big problem (making this approach less usefull and less mused than it could have been otherwise), is that, the approximation might be negative in some intervals, while a probability density function cannot be negative.

A quick thought assures us that two positive functions cannot be orthogonal, at lest not for the usual inner products of type $\int f(x) g(x) w(x) d\;x$.

So a natural idea is to ask for (non-orthogonal) expansions where each term in the expansion is non-negative. Does anybody know of any work in this direction?

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Aren't positive functions with non-overlapping support orthogonal?

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I don't think they can appear as eigenfunctions of the underlying Sturm-Liouville problem, because said functions have a finite amount of zeros. – Olivier Bégassat May 6 '11 at 3:07
Who said anything about an underlying Sturm-Liouville problem? – Robert Israel May 20 '11 at 1:46

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