Convolution-Composition-Multiplication

[markraz]

Hate to Hijack but since the question is answered, I have a question. Is this (f ∘ g) called Convolution?

 

[skeeter]

Hate to Hijack but since the question is answered, I have a question. Is this (f ∘ g) called Convolution?

Convolution -- from Wolfram MathWorld

A convolution is an integral that expresses the amount of overlap of one function g as it is shifted over another function f. It therefore "blends" one function with another. For example, in synthesis imaging, the measured dirty map is a convolution of the "true" CLEAN map with the dirty beam...

mathworld.wolfram.com

 

[JeffM]

Hate to Hijack but since the question is answered, I have a question. Is this (f ∘ g) called Convolution?

It is called a composition of functions and is a crucial concept in calculus.

 

[markraz]

Convolution -- from Wolfram MathWorld

A convolution is an integral that expresses the amount of overlap of one function g as it is shifted over another function f. It therefore "blends" one function with another. For example, in synthesis imaging, the measured dirty map is a convolution of the "true" CLEAN map with the dirty beam...

mathworld.wolfram.com

Thanks so what does this * mean??

 

[skeeter]

[MATH][f \ast g][/MATH]
looks like an asterisk to me ...

 

[Otis]

… what does [the asterisk] mean [ in (f*g)(t) ]?

Hi markraz. The asterisk means multiplication. Starting with Algebra, we stop using symbol × as a multiplication operator because it looks too much like the variable x. (Scientific Notation is one exception.)

(f * g)(t) = f(t) * g(t)

When composing functions (i.e., using the output of one function as the input to another), we use an open circle, instead.

(f ◦ g)(t) = f( g(t) )

?

 

[HallsofIvy]

Here, not just the "*" but the entire "[f*g]" is the "convolution" which is defined by the right side of the equation, \(\displaystyle \int_

It is called a composition of functions and is a crucial concept in calculus.

No, it is not! The composition of function is given by \(\displaystyle f(g(x))\).The "convolution of functions" is given by \(\displaystyle f\circ g= \int f(\tau)g(t- \tau)d\tau\).\)

 

[markraz]

Multiplication? Composition? which is it?

 

[skeeter]

[MATH](f \ast g)[/MATH] ... convolution, according to Wolfram Mathworld

[MATH](f \circ g)[/MATH] ... composition

[MATH](f \cdot g)[/MATH] ... multiplication