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I'm reading page 8 of Schaum's ouTlines "Signals and Systems" by Hwei P. HSU, and I'm not understanding where this formula for the nth generalized derivative (of a generalized function) comes from:

If $g(t)$ is a generalized function, its $n$th generalized derivative $g^{(n)}(t) = d^ng(t)/dt^n$ is defined by the following relation:

$$\int_{-\infty}^{\infty} \phi(t)g^{(n)}(t)dt = (-1)^n\int_{-\infty}^{\infty}\phi^{(n)}(t)g(t)dt$$

where $\phi(t)$ is a testing function which can be differentiated an arbitrary number of times and vanishes outside some fixed interval and $\phi^{(n)}(t)$ is the $n$th derivative of $\phi(t)$.

Where does this come from? I can't wrap my head into deriving it, nor can I find anywhere on the internet of how it is derived. This is not calculus!

Andres Kiani
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1 Answers1

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For a tempered distribution $g$ let $$g_k(x) = <g,k e^{-\pi k^2 (t-x)^2}>$$

$g_k$ is $ C^\infty$ with at most polynomial growth and for all $\phi $ Schwartz $$\lim_{k \to \infty}<g_k, \phi>=\lim_{k \to \infty}<g,k e^{-\pi k^2 t^2}\ast \phi>= <g,\phi>$$ thus $g_k\to g$ in the sense of distributions.

The $n$-th distributional derivative $g^{(n)}$ is just the limit in the sense of distributions of $g_k^{(n)}$ $$<g^{(n)},\phi> = \lim_{k \to \infty} <g_k^{(n)},\phi>= \lim_{k \to \infty}\int_{-\infty}^\infty g_k^{(n)}(t)\phi(t)dt$$

(integration by parts $n$ times)$$= \lim_{k \to \infty}(-1)^n\int_{-\infty}^\infty g_k(t)\phi^{(n)}(t)dt= \lim_{k \to \infty} (-1)^n<g_k,\phi^{(n)}>=(-1)^n <g,\phi^{(n)}>$$

reuns
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