deletions | additions       diff --git a/DeclareMathOperator_argmin_argmin_newcommand_reals__.tex b/DeclareMathOperator_argmin_argmin_newcommand_reals__.tex  index 483ec62..3d97fd8 100644  --- a/DeclareMathOperator_argmin_argmin_newcommand_reals__.tex  +++ b/DeclareMathOperator_argmin_argmin_newcommand_reals__.tex 
...
\begin{equation}  f(m) = \| \Fu m - y\|^2_{\ell_2} + \lambda \| \Psi m\|_{\ell_1}.  \end{equation}  The 'complex' derivative of the cost function $f$ at $m$, denoted $Df(m)$, is given by  \begin{equation}  Df(m) = D\| \Fu m - y\|^2_{\ell_2} + \lambda D\| \Psi m\|_{\ell_1}.  \end{equation}  Let's look at each term separately. For the first term, $D\| \Fu m - y\|^2_{\ell_2}$,   we apply the chain rule to the composite function $h(\Fu m - y)$,  where $h(x) = \|x\|^2_{\ell_2} : \complex^\nv \to \complex^\nk$.  It follows that  \ \begin{eqnarray}  Dh(\Fu m - y) &=& Dh(\fu m - y) D(\Fu m - y) \nonumber\\  &=& 2(\fu m - y)^T \Fu. \nonumber  \end{eqnarray}