Functional

Functional refers to a mapping from a space $\mathit{X}$ (usually of functions) into $\boldsymbol{R}$ (real numbers). (Or complex numbers, in a more general case.)

Generalized Function

Definition

Let \(\varphi(x)\) be an argument of the function, which is a function where $x \in \boldsymbol{R}$, and $\forall x$, $\varphi(x) \in \mathscr{D}$.

• $\varphi(x)$ is smooth, if it has derivatives of all orders.

• $\varphi(x)$ is compact, if it has a bounded support. The support of a function $\varphi(x)$ is written as $\text{supp}(\varphi)$, $\text{supp}(\varphi) := \{x \in \mathit{X} \mid \varphi(x) \ne 0 \}$.

Let $f(x) := \boldsymbol{R} \rightarrow \boldsymbol{R}$ be the kernel of $\varphi(x)$.

We have the following linear functional:

\[T[\varphi] = \int f(x) \varphi(x) dx\]

• $T[\cdot]$ is continuous, if for any sequence $\{\varphi_k\}_{k = 1, \cdots, \infty}$, when the sequence converges to $\varphi$ at $k \rightarrow \infty$, the corresponding sequence $\{T[\varphi_k]\}$ converges to $T[\varphi]$.

• $T[\cdot]$ maps inputs from $\mathscr{D}$ (function space) to $\boldsymbol{R}$.

Any linear functional $T[\varphi]$, which is continuous on the set $\mathscr{D}$ of smooth compact functions, is called a generalized function. A generalized function is a linear functional, but not a function.

Linearity of Generalized Function

$\forall \varphi(x), \psi(x) \in \mathscr{D}$, $\forall a, b \in \boldsymbol{R}$, we have:

\[T[a\varphi + b\psi] = a T[\varphi] + b T[\psi]\]

Regular Generalized Function

If the kernal $f(x)$ is an everywhere continuous, bounded function, $T[\varphi]$ is called a regular generalized function identified with the kernal $f(x)$.

Singular Generalized Function

A singular generlized function $T[\varphi]$ is still linear continuous, but its kernel is not continuous.

The most important example is the function:

\[T[\varphi] = \varphi(0)\]

Though the kernel is not continuous, we still give it a symbol $\delta(x)$ and write it as:

\[T[\varphi] = \int \delta(x) \varphi(x) dx = \varphi(0)\]

Although the above thing is well known as the delta function, according to Wikipedia and Wolfram, given different contexts, the delta function can refer to different things. As a distribution, the delta function is $T[\cdot]$, the linear functional, and is written as $\delta[\varphi] = \varphi(0)$. As a measure, or in the engineering context, the kernel $\delta(x)$ in $\int \delta(x) \varphi(x) dx$ is the delta function. A very confusing expression is:

\[\delta [\varphi] = \int \delta(x) \varphi(x) dx = \varphi(0)\]

If $\varphi(x) = 1$:

\[\delta [1] = \int \delta(x) dx = 1\]

We can also “shift” the delta function to map the function space to function value at arbitrary points: (again the first $\delta_a$ is a functional delta, the second $\delta$ in the integral is a function delta)

\[\delta_a [\varphi] = \int \delta(x - a) \varphi(x) dx = \varphi(a)\]

One approach to estimate the delta function is by the limit of a sequence of ordinary integrals, where the $k$ th element in the sequence looks like:

\[\delta_k [\varphi] = \int \delta_k(x) \varphi(x) dx\]

$\delta_k(x)$ can be ordinary functions. One choice is the Gaussian probability density function: $\delta_k(x) = \frac{1}{k\sqrt{2 \pi}} e^{-\frac{x^2}{2k^2}}$

Derivatives of Generalized Function

We define the derivative of a generalized function by the derivative of its kernel functions:

\[T'[\varphi] = \int f'(x) \varphi(x) dx\]

According to the rule of integration by parts, we have:

\[\int f'(x) \varphi(x) dx + \int f(x) \varphi'(x) dx = f(x)\varphi(x) |_{-\infty}^{\infty}\]

Recall that $\varphi(x)$ is compact (has a bounded support), so $\lim_{x \rightarrow \infty}\varphi(x) = 0$, $\lim_{x \rightarrow -\infty}\varphi(x) = 0$, therefore we have:

\[\int f'(x) \varphi(x) dx + \int f(x) \varphi'(x) dx = 0\]

Which means:

\[T'[\varphi] + T[\varphi'] = 0\]

So We can do derivative on $\varphi$ to get the derivative of a generalized function.

With the same logic, we have

\[T^{(n)}[\varphi] = (-1)^nT[\varphi^{(n)}(x)]\]

The Leibniz formula also works on the product of a generalized function and a function. First we have the product of a function and a generalized function defined as:

\[g(x)T[\varphi] = T[g\varphi]\]

Then the Leibniz formula is:

\[(g(x)T[\varphi])' = g'(x)T[\varphi] + g(x)T'[\varphi]\] \[(g(x)T[\varphi])^{(n)} = \sum^{n}_{m = 0}g^{(m)}(x)T^{(n-m)}[\varphi]\]

This is more like a trick, since the symbol $g’(x)T[\varphi]$ is not an actual product, you just switch between different expressions to make it look like the Leibniz formula :

\[\begin{align*} (g(x)T[\varphi])' &= T'[g\varphi] \\ &= T[(g\varphi)'] \\ &= T[g'\varphi + g\varphi'] \\ &= T[g'\varphi] + T[g\varphi'] \text{ (linearity)} \\ &= g'(x)T[\varphi] + g(x)T'[\varphi] \end{align*}\]

Generalized Function of a Composite Argument

First we have the following equality:

\[\int f(g(x))\varphi(x) dx = \int f(y)\varphi(g^{-1}(y)) g^{-1 \prime} (y) dy\]

Where $g^{-1}(x)$ is the inverse of $g(x)$, i.e. $g(g^{-1}(x)) = x$. The above equality can be easily proved by replace $x$ on the left side with $g^{-1}(y)$.

We name $\int f(g(x))\varphi(x) dx$ as a generalized function $T$ of a composite argument $g(x)$.