The Fourier Transform
The Fourier transform is one of the most productive tools of the mathematical sciences. It crops up again and again in unexpected applications to fields as diverse as differential equations, numerical analysis, probability theory, number theory, quantum mechanics, optics, medical imaging, and signal processing [3, 5, 7, 8, 9]. One explanation for its wide utility is that it turns complex mathematical operations like differentiation and convolution into simple operations like multiplication. Readers most likely are familiar with the paradigm of transforming a mathematical equation, solving it in transform space, and then inverting the solution. Besides its operational advantages, the Fourier transform often has the illuminating physical interpretation of decomposing a temporal process into component processes with different frequencies.
KeywordsFourier Transform Saddle Point Integrable Function Moment Generate Function Saddle Point Approximation
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