Hi, Does anyone know the necessary and sufficient conditions for a function to be a DC-function?
Definition: A function is a DC-function if and only if it can be written as a differnece of 2 convex functions.
1 Answer
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For real functions whose domain is a real interval, it is necessary and sufficient that the second derivative is a function of bounded variation on every compact interval in the domain. Or, in terms of distributions, the second derivative must be a measure (a difference of two non-negative measures).
answered Nov 14, 2012 at 22:59
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1$\begingroup$ This is for real-to-real function only. $\endgroup$ Nov 14, 2012 at 23:11
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$\begingroup$ $x \: \mapsto \: x^4-0 \;\;$ seems like a counterexample to the necessity. $\;\;\;\;$ $\endgroup$– user5810Nov 15, 2012 at 7:47
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$\begingroup$ Yes, this was for functions R to R. Concerning the counterexample, I edited the statement:-) $\endgroup$ Nov 15, 2012 at 12:49