nLab generalized function

Idea

In functional analysis, distributional densities are usefully thought of as generalized smooth functions in that every distribution is the limit of a sequence of smooth functions, thought of as non-singular distributions (this prop.).

More in detail, regarding a smooth function $f \in C^\infty(X)$ as a distributional density means to regard it as the integral kernel

$$u_f \;\colon\; b \mapsto \underset{X}{\textstyle{\int}} f(x) b(x) \, dvol_X(x)$$

on bump functions $b \in C^\infty_{cp}(X)$. Therefore a limit over a sequence $\{f_n\}_{n \in \mathbb{N}}$ of smooth functions, which may not really exists as a smooth function but just as a distribition, is nevertheless usefully thought of as the integral against the “generalized function” which would be “$\underset{\underset{n}{\longrightarrow}}{\lim} f_n$” if that limit existed in $C^\infty(X)$:

$$\underset{X}{\textstyle{\int}} \left(\underset{\underset{n}{\longrightarrow}}{\lim} f_n\right)(x) b(x) \, dvol_X(x) \;\coloneqq\; \underset{\underset{n}{\longrightarrow}}{\lim} \underset{X}{\textstyle{\int}} f_n(x) b(x), dvol_X(x) \,.$$

It is common and convenient to write evaluation of distributions on test functions in this way, and the name of many distributions reflects this habit, such as “Dirac delta function” for the delta distribution or “commutator function” for a causal propagator.

In particular distributions that arise from integral expressions appearing in Fourier analysis are often conveniently written this way: For $f \in \mathbb{R}^n$ a smooth function (or distribution) in a variable $k$, its Fourier transform of distributions $\hat f$ generally is given as a generalized function by the integral expression

$$\hat f(x) \;=\; \underset{\mathbb{R}^n}{\textstyle{\int}} f(k) e^{i k \cdot x} \, d^n k$$

which however in general is not an actual function of $x$, in that this integral need not converge for fixed $x \in \mathbb{R}^n$, but is a generalized function in that for appropriate test functions $b$ the further integral

$$\underset{\mathbb{R}^n}{\textstyle{\int}} \underset{\mathbb{R}^n}{\textstyle{\int}} f(k) e^{i k \cdot x} b(x) \, d^n x \, d^n k \;\in\; \mathbb{C}$$

does make sense.

For example it is in this sense that the delta-distribution, regarded as a generalized function, is given by the common integral expression

$$\delta(x) \;=\; \underset{k \in \mathbb{R}^n}{\textstyle{\int}} e^{2 \pi i k x} d^n k$$

(see at delta distribution this example).

Thinking of distributions as generalized functions this way motivates and streamlines the extension of a variety of concepts from functions to distributions, such as the concepts of product of distributions, derivative of distributions, support of distributions, distributional solutions to a PDE and so on.

Related concepts

• distribution

• hyperfunction

• current (distribution theory)