Chapter 10. Fourier Transforms and the Dirac Delta Function

Vector Spaces in Physics 8/6/2015 10 - 1 Chapter 10. Fourier Transforms and the Dirac Delta Function A. The Fourier transform . The Fourier - series expansions which we have discussed are valid for functions either defined over a finite range (/ 2/ 2Tt T , for instance) or extended to all values of time as a periodic Function . This does not cover the important case of a single, isolated pulse. But we can approximate an isolated pulse by letting the boundaries of the region of the Fourier series recede farther and farther away towards , as shown in figure 10-1. We will now outline the corresponding mathematical limiting process. It will transform the Fourier series , a superposition of sinusoidal waves with discrete frequencies n, into a superposition of a continuous spectrum of frequencies.

as the Fourier series is an expansion in terms of a series of orthogonal functions. Here is the picture. Basis states The functions e i t 2 1 Ö( ) . (10-21) constitute a complete orthonormal basis for the space of ''smooth'' functions on the interval t . We are not going to prove completeness; as with the Fourier series, the fact that the

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  Series, Chapter, Functions, Chapter 10, Delta, Transform, Fourier, Fourier series, Orthogonal, Carid, Orthogonal functions, Fourier transforms and the dirac delta

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