What Does It Mean for a Function to Have Compact Support?

In summary, "compact support" means that the function is zero everywhere outside a finite interval and vanishes at positive and negative infinity. The Gaussian function does not have compact support.
  • #1
mnb96
715
5
Hello,
given a function f:R->R, can anyone explain what is meant when we say that "f has compact support"?

Some sources seem to suggest that it means that f is non-zero only on a closed subset of R.
Other sources say that f vanishes at infinity. This definition seem to contradict the previous: for example the Gaussian is never 0 but does vanish at infinity.

So, where is the misunderstanding?
 
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  • #2
Compact support means the function is zero everywhere outside some finite interval. Gaussian does not have compact support.
 
Last edited:
  • #3
It naturally means that the support of the function is a compact set, or equivalently as mathman points out; contained in a finite closed interval. This implies that f must vanish at positive and negative infinity, but is not equivalent as your example shows.
 

Related to What Does It Mean for a Function to Have Compact Support?

What is the compact support of a function?

The compact support of a function is the subset of its domain where the function is non-zero. In other words, it is the set of points where the function is defined and has a finite value.

Why is it important to consider the compact support of a function?

The compact support of a function is important because it allows us to restrict our attention to a finite region, which can make certain calculations and proofs easier. Additionally, functions with compact support have many useful properties, such as being continuous and having a well-defined integral.

How is the compact support of a function related to its domain and range?

The compact support of a function is a subset of its domain, and it determines the range of the function. If a function has compact support, then its range is limited to a finite region.

What is the difference between a function with compact support and one without?

A function with compact support is defined and has a finite value only within a specific region, while a function without compact support can have non-zero values in an infinite region. In other words, the compact support of a function restricts its domain and range.

Are there any real-world applications of compact support functions?

Yes, compact support functions are commonly used in signal processing, where the functions represent signals that are limited in time or space. They are also used in physics, particularly in quantum mechanics, to represent physical systems with finite energy or position.

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