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7. Extensions of the Fourier transform concept
The Fourier transform has proved so useful, in so many applications, that many mathematicians have generalized it, or described other transformations with similar properties. In this section, we describe some of them.
7.1 Laplace transformation
This transformation actually predates Fourier's in historical terms. It was used by Laplace for real functions in 1812; its inverse was explained by Poisson in 1820. It was subsequently generalized and, today, the Fourier transform can be considered a special case. However, the Laplace transform of distributions cannot be defined so simply. The following is a brief overview of its various characteristics, without proof or details.
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Extensions of the Fourier transform concept
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