Some approximation properties in fractional Musielak–Sobolev spaces

In this article we show some density properties of smooth and compactly supported functions in fractional Musielak–Sobolev spaces essentially extending the results of Fiscella et al. (Ann Acad Sci Fenn Math 40(1):235–253, 2015) obtained in the fractional Sobolev setting. The proofs of these properties are mainly based on a basic technique of convolution (which makes functions \(C^{\infty }\) $C^{\infty }$ ), joined with a cut-off (which makes their support compact), with some care needed in order not to exceed the original support.