nLab rapidly decreasing function

Idea

A function on a Cartesian space is called rapidly decreasing if every the product with any power of the canonical coordinate functions is a bounded function (def. below).

Of particular interest are the smooth functions with are rapidly decreasing and all whose partial derivatives are rapidly decreasing, too (def. below). These Schwartz functions enjoy the special property that the operation of Fourier transform is an endomorphism on the Schwartz space of all these functions. This gives them a central place in harmonic analysis. A tempered distribution is a continuous linear functional on this Schwartz space.

Definition

Rapidly decreasing functions

Definition

(rapidly decreasing function)

For $n \in \mathbb{N}$, an integrable function

$$f \colon \mathbb{R}^n \to \mathbb{R}$$

on a Cartesian space $\mathbb{R}^n$ is called rapidly decreasing if for all $\alpha \in \mathbb{N}^n$ the product function

$$x \mapsto x^\alpha f(x)$$

is a bounded function. Here

$$x^\alpha \;\coloneqq\; (x^1)^{\alpha_1} \cdots (x^n)^{\alpha_n}$$

is any given product of powers of the canonical coordinate functions.

Functions with rapidly decreasing partial derivatives

Definition

(function with rapidly decreasing partial derivatives)

A smooth function

$$f \;\colon\; \mathbb{R}^n \longrightarrow \mathbb{R}$$

is a function with rapidly decreasing partial derivatives or Schwartz function for short, if all its partial derivatives are rapidly decreasing functions in the sense of def. , hence if for all $\alphe, \beta \in \mathbb{N}^n$ we have that

$$x \mapsto x^\alpha \partial_{\beta} f$$

is a bounded function, where

$$\partial_\beta \;\coloneqq\; \frac{\partial^{\beta_1}}{\partial (x^1)^{\beta_1}} \cdots \frac{\partial^{\beta_n}}{\partial (x^n)^{\beta_n}}$$

are the given partial derivatives.

Remark

These Schwartz functions (def. ) form the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ used in the definition of tempered distributions in functional analysis.

Properties

Proposition

(rapidly decreasing functions are integrable functions)

If $f \colon \mathbb{R}^n \longrightarrow \mathbb{R}$ is a rapidly decreasing function, then its integral exists

$$\int_{x \in \mathbb{R}^n} f(x) \, dvol(x) \;\lt \; \infty \,.$$

In fact for all $\alpha \in \mathbb{N}^n$ the integral of $x \mapsto x^\alpha f(x)$ exists:

$$\int_{x \in \mathbb{R}^n} x^\alpha f(x) \, dvol(x) \;\lt \; \infty \,.$$

Examples

Example

(compactly supported smooth funtions? are functions with rapidly decreasing partial derivatives)

Every compactly supported smooth function (bump function) $b \in C^\infty_{cp}(\mathbb{R}^n)$ is rapidly decreasing (def. ) has rapidly decreasing partial derivatives (def. ):

$$C^\infty(\mathbb{R}^n) \hookrightarrow \mathcal{S}(\mathbb{R}^n) \,.$$

(Non-)Example

Not every rapidly decreasing function (def. ) has rapidly decreasing partial derivatives (def. ).

For example the function $f \colon \mathbb{R} \to \mathbb{R}$

$$f(x) \coloneqq e^{-x^2}\sin(e^{x^2})$$

is rapidly decreasing, but its first derivative

$$f'(x) = -2xe^{-x^2}\sin(e^{x^2})+2x\cos(e^{x^2})$$

is asymptotically linearly increasing, due to the second term.

Related concepts

• continuous function, differentiable function, smooth function

• integrable function, square-integrable function

• bounded function

• compactly supported function

• measurable function