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Distributions and weak limits: The shadow of a function

Much as Plato described the effect of the ideal ``forms'' in terms of their shadows, modern mathematicians often desribe the functions in terms of the effect on other functions. Distributions deal with the limits of the type (2). It becomes easy to deal with some limits as the $\delta$-function whose influence on a trial function $\psi$ is quite simple:

\lim_{n\rightarrow \infty}\int_{- \infty}^{\infty} g_n(x) \, \psi(x)\,
dx = \psi(0).

The next example of infinitely often wiggling curve in 4.3.3 shows that sometimes weak limits are not sensitive enough to desribe strange limits.

Andre Cherkaev