How to typeset a small black square as a binary operator?

This is actually not a square but just the shape of \cdot from the Lucida family used in the picture.

\documentclass{article}
\usepackage{lucidabr}
\begin{document}
$f \cdot g$
\end{document}

Lucida is a commercial font, you can purchase it from TUG.

---

You could also go ahead and fake the symbol using a \rule.

\documentclass{article}
\renewcommand\bullet{%
  \mathbin{\mskip1mu
    \mathchoice
      {\squarebullet{.25ex}{.25ex}}%
      {\squarebullet{.25ex}{.25ex}}%
      {\squarebullet{.18ex}{.18ex}}%
      {\squarebullet{.15ex}{.15ex}}
    \mskip1mu}
}
\newcommand\squarebullet[2]{\vcenter{\hbox{\rule{#1}{#2}}}}
\begin{document}
$f \cdot g_{f \cdot g_{f \cdot g}}$

$f \bullet g_{f \bullet g_{f \bullet g}}$
\end{document}

This symbol exists in the mathabx font package. Here is a way to use this symbol without having to load the package:

\documentclass{article}
\DeclareFontFamily{U}{mathb}{\hyphenchar\font45}
\DeclareFontShape{U}{mathb}{m}{n}%
{<-6> mathb5 %
 <6-7> mathb6
 <7-8> mathb7 %
 <8-9> mathb8 %
 <9-10> mathb9 %
 <10-12> mathb10 %
 <12-> mathb12 }%
 {}
\DeclareSymbolFont{mathb}{U}{mathb}{m}{n}
\DeclareMathSymbol{\sqbullet}{\mathbin}{mathb}{"0D}

\begin{document}

\[ f \sqbullet g\]%

\end{document} 

I don't know the exact size of the black square as in the picture of the OP. I have made this with a new command \bsq:

\documentclass[12pt]{article}
\usepackage{amssymb,amsmath}
\usepackage{scalerel}
\newcommand{\bsq}{{\mkern 5mu}\blacksquare{\mkern 9mu}}

\begin{document}
\[f\raisebox{1pt}{\scaleto{\bsq}{2pt}} g(s)=
\begin{cases}
f(2s) &\\
g(2s-1)&
\end{cases}\]

\end{document}