  Homogeneous mixed-norm Triebel--Lizorkin spaces are introduced and studied
with the use of a discrete wavelet transformation, the so-called
$\varphi$-transform. This extends the classical $\varphi$-transform approach
introduced by Frazier and Jawerth to the setting of mixed-norm spaces.
Moreover, the theory of the $\varphi$-transform is enhanced through a precise
definition of the synthesis operator, in terms of a Pettis integral, and a
number of rigorous results for this operator. Especially its terms can always
be summed in any order, without changing the resulting distribution.
