What is a Functional Transformation?

Date: 08/11/2002 at 00:17:37
From: Britney
Subject: Definition of a functional transformation

Hi Dr. Math,

I would like a brief definition of what a functional transformation is 
and what an application is in early Calculus. 

Thank you in advance.

Date: 08/11/2002 at 16:14:49
From: Doctor Jaime
Subject: Re: Definition of a functional transformation

Hello,

A "functional transformation" or a "functional operator" is just a map 
(function) whose domain is some set of functions (e.g. all functions 
defined in the interval [0,2], all continuous functions, all real 
functions with the same domain D). The range can be a set of numbers, 
a set of functions or some other set.

So the difference between a "functional transformation" and maps you 
have already studied is that the domain is a bit more complicated: 
functions acting on functions.

Confused? You have already studied a "functional transformation" but 
never gave it that name: the derivative. In fact a derivative is 
applied to functions and the result is another function, the 
derivative function. That's why this operation is usually called the
"differential operator":

   Differential Operator - MathWorld
   http://mathworld.wolfram.com/DifferentialOperator.html 

I don't know if you have already studied integrals, but the definite 
integral, the indefinite integral and the improper integral are also 
examples of functional transformations, usually called "integral 
transforms":

   Integral Transform - MathWorld
   http://mathworld.wolfram.com/IntegralTransform.html 

You can study properties of these functional transformations, like 
linearity. If they are linear they are called "linear 
transformations":

   Linear Transformation - MathWorld
   http://mathworld.wolfram.com/LinearTransformation.html 

Some famous examples of functional transformation are the Laplace 
transformation and the Fourier transformation, which are intensively 
used in physics and engineering. You can read something about them 
here:

   Laplace Transform - MathWorld
   http://mathworld.wolfram.com/LaplaceTransform.html 

   Fourier Transform - MathWorld
   http://mathworld.wolfram.com/FourierTransform.html 

Please feel free to ask again if you have more questions. And don't 
forget to visit our FAQ at

   http://mathforum.com/dr.math/faq/ 

- Doctor Jaime, The Math Forum
  http://mathforum.org/dr.math/