Reduced Derivative - Definition

Definition

Let X be a separable, reflexive Banach space with norm || || and fix T > 0. Let BV₋(X) denote the space of all left-continuous functions z : → X with bounded variation on .

For any function of time f, use subscripts +/− to denote the right/left continuous versions of f, i.e.

For any sub-interval of, let Var(z, ) denote the variation of z over, i.e., the supremum

The first step in the construction of the reduced derivative is the “stretch” time so that z can be linearly interpolated at its jump points. To this end, define

The “stretched time” function τ̂ is left-continuous (i.e. τ̂ = τ̂₋); moreover, τ̂₋ and τ̂₊ are strictly increasing and agree except at the (at most countable) jump points of z. Setting T̂ = τ̂(T), this “stretch” can be inverted by

Using this, the stretched version of z is defined by

where θ ∈ and

The effect of this definition is to create a new function ẑ which “stretches out” the jumps of z by linear interpolation. A quick calculation shows that ẑ is not just continuous, but also lies in a Sobolev space:

The derivative of ẑ(τ) with respect to τ is defined almost everywhere with respect to Lebesgue measure. The reduced derivative of z is the pull-back of this derivative by the stretching function τ̂ : → . In other words,

Associated with this pull-back of the derivative is the pull-back of Lebesgue measure on, which defines the differential measure μ_z: