Wiener–Ikehara theorem

In mathematics, the metric derivative is a notion of derivative appropriate to parametrized paths in metric spaces. It generalizes the notion of "speed" or "absolute velocity" to spaces which have a notion of distance (i.e. metric spaces) but not direction (such as vector spaces).

Definition

Let $(M,d)$ be a metric space. Let $E\subseteq \mathbb {R}$ have a limit point at $t\in \mathbb {R}$. Let $\gamma :E\to M$ be a path. Then the metric derivative of $\gamma$ at $t$, denoted $|\gamma '|(t)$, is defined by

$|\gamma '|(t):=\lim _{s\to 0}{\frac {d(\gamma (t+s),\gamma (t))}{|s|}},$

if this limit exists.

Properties

Recall that AC^p(I; X) is the space of curves γ : I → X such that

$d\left(\gamma (s),\gamma (t)\right)\leq \int _{s}^{t}m(\tau )\,\mathrm {d} \tau {\mbox{ for all }}[s,t]\subseteq I$

for some m in the L^p space L^p(I; R). For γ ∈ AC^p(I; X), the metric derivative of γ exists for Lebesgue-almost all times in I, and the metric derivative is the smallest m ∈ L^p(I; R) such that the above inequality holds.

If Euclidean space $\mathbb {R} ^{n}$ is equipped with its usual Euclidean norm $\|-\|$, and ${\dot {\gamma }}:E\to V^{*}$ is the usual Fréchet derivative with respect to time, then

$|\gamma '|(t)=\|{\dot {\gamma }}(t)\|,$

where $d(x,y):=\|x-y\|$ is the Euclidean metric.

References

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