If we have a function f that has a minimum at a point say x. Does f preserve the min if we convolve it with a mollifier?
closed as too localized by Harald Hanche-Olsen, Willie Wong, Deane Yang, S. Carnahan♦ Nov 8 '10 at 2:24
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No, consider a function $f$ with a graph something like the above for example. (Pardon the rough sketch:-) When you mollify it the minimum at the origin will rise and become greater that the value of the mollified function away from the origin.