rapidly decreasing function

**analysis** (differential/integral calculus, functional analysis, topology)

metric space, normed vector space

open ball, open subset, neighbourhood

convergence, limit of a sequence

compactness, sequential compactness

continuous metric space valued function on compact metric space is uniformly continuous

…

…

A function on a Cartesian space is called *rapidly decreasig* if every the product with any power of the canonical coordinate functions is a bounded function (def. 1 below).

Of particular interest are the smooth functions with are rapidly decreasing and all whose partial derivatives are rapidly decreasing, too (def. 2 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.

**(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.

**(function with rapidly decreasing partial derivatives)**

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

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. 1, 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.

These Schwartz functions (def. 2) form the *Schwartz space* $\mathcal{S}(\mathbb{R}^n)$ used in the definition of tempered distributions in functional analysis.

**(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:

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

**(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. 1) has rapidly decreasing partial derivatives (def. 2):

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

Not every rapidly decreasing function (def. 1) has rapidly decreasing partial derivatives (def. 2).

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.

Last revised on November 7, 2017 at 11:12:41. See the history of this page for a list of all contributions to it.