Principles of Fourier Analysis, 2nd Edition

Relation with Classical Derivatives

Let’s see how f′ is related to Df when f is a piecewise smooth function whose classical derivative, f′, is exponentially integrable. For now, assume f has exactly one discontinuity, say, a jump of j₀ at x₀. Remember,

$j_0=\underset{x\to x_0^{+}}{\lim }f\left(x\right)-\underset{x\to x_0^{-}}{\lim }f\left(x\right).$

At this point we should observe that f(x) can be computed from its derivative via

$f\left(x\right)-\underset{s\to x_0^{+}}{\lim }f\left(s\right)={\displaystyle {\int }_{x_0}^{x}f\left(s\right)ds}\text{when}{x}_0<x$

and

$\underset{s\to x_0^{-}}{\lim }f\left(s\right)-f\left(x\right)={\displaystyle {\int }_x^{x_0}f\prime \left(s\right)ds}\text{when}x<x_0.$

From this and the exponential integrability of f′ we can easily deduce that f, itself, must be exponentially bounded on the real line (see lemma 30.10 on page 496).

Now let ϕ be any Gaussian test function. By definition,

$\langle Df,\varphi \rangle =-\langle f,\varphi \prime \rangle =-{\displaystyle {\int }_{-\infty }^{\infty }f\left(x\right)\varphi \prime \left(x\right)dx}.$

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