  In this paper we consider the minimization of a novel class of fractional
linear growth functionals involving the Riesz fractional gradient. These
functionals lack the coercivity properties in the fractional Sobolev spaces
needed to apply the direct method. We therefore utilize the recently introduced
spaces of bounded fractional variation and study the extension of the linear
growth functional to these spaces through relaxation with respect to the weak*
convergence. Our main result establishes an explicit representation for this
relaxation, which includes an integral term accounting for the singular part of
the fractional variation and features the quasiconvex envelope of the
integrand. The role of quasiconvexity in this fractional framework is explained
by a technique to switch between the fractional and classical settings. We
complement the relaxation result with an existence theory for minimizers of the
extended functional.
