Pure mathematics is much more than an armory of tools and techniques for the applied mathematician. On the other hand, the pure mathematician has ever been grateful to applied mathematics for stimulus and inspiration. From the vibrations of the violin string they have drawn enchanting harmonies of Fourier Series, and to study the triode valve they have invented a whole theory of non-linear oscillations.
George Frederick James Temple In 100 Years of Mathematics: a Personal Viewpoint (1981).
The Fourier Transform (FT) is an integral transform, a powerful mathematical tool to map a function from its original space representation into another function space (called, in this case, the Fourier space). In the time domain, the Fourier space is the frequency and in the Cartesian domain is the so-called reciprocal space. The FT is accomplished by integrating the given function in its original space. The advantage of the FT is that in the transformed space, the properties of the original function can usually be characterised and manipulated more quickly than in the original function space. The FT function can generally be mapped back to the original function space using the inverse FT.
The FT plays an important role in pure and applied science, computer science, electronic engineering, and medicine. In this lecture, I will shortly introduce the mathematics of the FT and then show some examples of practical applications.
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