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/ |
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