  This paper concerns the universality of the two-layer neural network with the
$k$-rectified linear unit activation function with $k=1,2,\ldots$ with a
suitable norm without any restriction on the shape of the domain. This type of
result is called global universality, which extends the previous result for
$k=1$ by the present authors. This paper covers $k$-sigmoidal functions as an
application of the fundamental result on $k$-rectified linear unit functions.
