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Consider two continuous linear operators $T\colon X_1(\mu)\to Y_1(\nu)$ and$S\colon X_2(\mu)\to Y_2(\nu)$ between Banach function spaces related todifferent $\sigma$-finite measures $\mu$ and $\nu$. We characterize by means ofweighted norm inequalities when $T$ can be strongly factored through $S$, thatis, when there exist functions $g$ and $h$ such that $T(f)=gS(hf)$ for all$f\in X_1(\mu)$. For the case of spaces with Schauder basis ourcharacterization can be improved, as we show when $S$ is for instance theFourier operator, or the Ces\`aro operator. Our aim is to study the case whenthe map $T$ is besides injective. Then we say that it is a~representingoperator ---in the sense that it allows to represent each elements of theBanach function space $X(\mu)$ by a~sequence of generalized Fouriercoefficients---, providing a complete characterization of these maps in termsof weighted norm inequalities. Some examples and applications involving recentresults on the Hausdorff-Young and the Hardy-Littlewood inequalities foroperators on weighted Banach function spaces are also provided.

Strong factorizations of operators with applications to Fourier and Ces\'aro transforms