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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


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 

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 

   Integral Transform - MathWorld 

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

   Linear Transformation - MathWorld 

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 

   Laplace Transform - MathWorld 

   Fourier Transform - MathWorld 

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

- Doctor Jaime, The Math Forum 
Associated Topics:
College Calculus
College Definitions
High School Calculus
High School Definitions
High School Functions

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