The inversion formula gives a uniqueness result which

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The inversion formula gives a uniqueness result which is often more useful than the inversion formula itself. 18
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Theorem 7.5 (Uniqueness) . If ˆ f = ˆ g then f = g . Combining the inversion formula with Lemma 7.3, we get the following formula which is much loved by 1B examiners and of considerable theoretical importance. Lemma 7.6 (Parseval’s formula) . 7 We have integraldisplay −∞ | f ( t ) | 2 dt = 1 2 π integraldisplay −∞ | ˆ f ( λ ) | 2 dλ. Fourier transforms are closely linked with the important operation of convolution. Definition 7.7. If f, g : R C are well behaved, we define their convolu- tion f g : i R C by f g ( t ) = integraldisplay −∞ f ( t s ) g ( s ) ds. Lemma 7.8. We have hatwide f g ( λ ) = ˆ f ( λ g ( λ ) . For many mathematicians and engineers, Fourier transforms are impor- tant because they convert convolution into multiplication and convolution is important because it is transformed by Fourier transforms into multiplica- tion. We shall see that convolutions occur naturally in the study of differ- ential equations. It also occurs in probability theory where the sum X + Y of two independent random variables X and Y with probability densities f X and f Y is f X + Y = f X f Y . In the next section we outline the connection of convolution with signal processing. 8 Signals and such like Suppose we have a black box K . If we feed in a signal f : R C we will get out a transformed signal K f : R C . Simple black boxes will have the following properties (1) Time invariance If T a f ( t ) = f ( t a ), then K ( T a f )( t ) = ( K f )( t a ). In other words, KT a = T a K . (2) Causality If f ( t ) = 0 for t < 0, then ( K f )( t ) = 0 for t < 0. (The response to a signal cannot precede the signal.) 7 The opera has an ‘f’ and goes on for longer. 19
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(3) Stability Roughly speaking, the black box should consume rather than produce energy. Roughly speaking, again, if there exists a R such that f ( t ) = 0 for | t | ≥ R , then we should have ( K f )( t ) 0 as t → ∞ . If conditions like this do not apply, both our mathematics and our black box have a tendency to explode. (Unstable systems may be investigated using a close relative of the Fourier transform called the Laplace transform.) (4) Linearity In order for the methods of this course to work, our black box must be linear, that is K ( af + bg ) = a K ( f ) + b K ( g ) . (Engineers sometimes spend a lot of effort converting non-linear systems to linear for precisely this reason.) As our first example of such a system, let us consider the differential equation F ′′ ( t ) + ( a + b ) F ( t ) + abF ( t ) = f ( t ) (where a, b > 0), subject to the boundary condition F ( t ) , F ( t ) 0 as t → −∞ . We take K f = F . Before we can solve the system using Fourier transforms we need a pre- liminary definition and lemma. Definition 8.1. The Heaviside function H : R R is given by H ( t ) = 0 for t< 0 , H ( t ) = 1 for t 0 .
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