Variation Seminorm

The -variation seminorm of a function on an ordered interval, for , is

The supremum is over all finite increasing partitions in the domain. It is a seminorm because adding a constant function does not change it. For on a compact interval, it agrees with the total variation of a function. For a monotonic function with real values on , for every .

When also depends on a spatial variable, the partition can be selected independently at every spatial point. Taking an L^p-space norm of that pointwise variation is stronger than placing the partition supremum outside the norm. Annular variation of the triangular Hilbert transform is an example.