This is the incontrovertible flaw in the death penalty. You can't undo it if the wrong man is executed. The fact that hundres of men have spent years, even decades, on death row, only to be exonerated by technologies undreamed of at the time of the crime indicates that shortening the period between conviction and sentence only increases the number of innocent men who will be killed.
Nobody does DNA studies (or any other reviews) on people after they have been executed. Not only is there no point to it, but the minimal resources for this kind of work are better spent on those who can still be saved, and, in many states, after the execution, the evidence is tossed in the garbage bin.
The most common cause of a wrongful conviction is mistaken (or suborned) eye-witness testimony. Every attorney knows how unreliable this is, but also knows that it flat-out sells the case to the jury. There was a case of a man who was convicted of rape on the eye-witness testimony of the victim - his cousin. She, of all people, should have been capable of making an accurate identification. But, after 8 years in jail, his DNA was finally compared to the DNA taken from her rape kit, and it turned out he didn't commit the rape.
The second most common cause of wrongful conviction is malfeasance on the part of the prosecution, which conceals exculpatory evidence, makes deals for shorter sentences with sleazeballs for testimony ("Members of the jury, this witness, 'Icepick' McGurk, killed three people over a $50.00 drug ripoff, but he would NEVER stoop to lying under oath, even to get his life sentence shortened.").
