MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question

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).

share|cite|improve this answer
1  
This is for real-to-real function only. – Anton Petrunin Nov 14 '12 at 23:11
    
$x \: \mapsto \: x^4-0 \;\;$ seems like a counterexample to the necessity. $\;\;\;\;$ – Ricky Demer Nov 15 '12 at 7:47
    
Yes, this was for functions R to R. Concerning the counterexample, I edited the statement:-) – Alexandre Eremenko Nov 15 '12 at 12:49

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.