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Limit of an infinitely fast wiggling curve

The limiting infinitely fast wiggling curve

\begin{displaymath}{\cal G}= \lim \sin( \omega x),\quad \omega \rightarrow \infty

is not a function either: Although the range of its values is $ [-1, 1]$, one cannot compute its value ${\cal G}(x)$ for any specific $x$ except $x=0 $.

\Huge {Animated figure
to be inserted here}

In addition, its average over any interval is zero, and its weak limit (see the discussion of the term in 4.3.2) is zero:

\lim_{\omega \rightarrow \infty} \int_{-1}^1 \sin( \omega x) \psi(x) dx =0,
\quad \forall \psi(x)\in L_\infty
\end{displaymath} (2)

However, some moments of ${\cal G}$ are nonzero. The second moment is:

\begin{displaymath}\lim_{\omega \rightarrow \infty} \int_{-1}^1 \sin^2( \omega x...
...ver 2} + {1 \over 2} \cos(2 \omega x) \right) dx
={1 \over 2}

Similarly, higher even moments are non-zero:

\lim_{\omega \rightarrow \infty} \int_{-1}^1 \sin^3( \omega x...
...rrow \infty} \int_{-1}^1 \sin^4( \omega x) dx & =&
{3 \over 8}

and so on.

Andre Cherkaev