As lawmakers on Capitol Hill revive talk of gun control following the school shooting in Newtown, Conn., a federal appeals court in Washington this week will take a look at the propriety of new firearm regulations that target the sale of semi-­automatic rifles.

Federal firearm authorities in July 2011 issued so-called “demand letters” to every gun dealer in Arizona, California, New Mexico and Texas, requiring thousands of retail sellers to report sales of more than one semi-automatic rifle to the same person at one time or during a period of five consecutive business days.