Functions have the ability to be described in terms of the infinite summation of other functions with a common example being polynomials using Taylor series as shown below.
However, functions can also be described in terms of trigonometric functions using the Fourier series as shown below.
Using complex exponentials, this can be rewritten.
is simply the amplitude of angular frequency . Notice this creates a definition for that can be imaginary but it can be kept completely real by making the property true or completely imaginary if . If one were to graph as a function of , a discrete function would be formed giving information on what frequencies have higher amplitudes in the function. A function like , for example, would have a graph of with four points at and . This would be equivalent to defining the equation as which, if simplified, is the equation presented earlier.
In order to obtain values for that are not discrete, the function must be defined in terms of some integral of frequencies instead of summation which is done below.
The integral can be seen here, in some sense, as a summation like the one presented earlier. represents the amplitude of any given frequency and is called the function’s Fourier transform. Applying this integral is called the inverse Fourier transform because it takes a known function in frequency space and creates the original function. Note that was changed to making the function a function of time. In theory, given just its Fourier transform, one could construct the original function as a function of . In fact, can be treated as a definition of the function itself with the only difference being that it is defined in frequency space instead of time space. One can however create the Fourier transform from the original function. There is actually some symmetry in the definitions of and as shown below.
This is called the forward Fourier transform. In this way, the two functions are actually very closely connected and acts as a very useful way to describe functions in a different way. These transforms actually become very useful in the analysis of concepts throughout physics, math, and computer science. In fact, for a function in position space, its Fourier transform will be a function in momentum space so there are very fundamental connections that can be made through this concept. The forward Fourier and inverse Fourier operators of a function are usually notated and respectively.
The Laplace transform is also important. A Laplace transform is very closely related to the Fourier transform as shown below.
Again, this transfers a function from time space to frequency space but in this transform, it is an exponential frequency space. It is used extensively in linear differential equations and especially in electrical circuits. This concept will be referred to many more times in this site in the future as well.
There are, in general, an immense amount of transforms that are used to put functions into spaces that are easier to understand. These are simply the two most common ones. Here is the list of some of the many: Wigner-Weyl, Mehler-Fokk, Gabor, Hermite, Berezin, Watson, Cayley, Meijer, Wihttaker, Penrose, Hardy, Kontorovich-Lebedev, Gauss, Olevskii, Jacobi, Laguerre, Hilbert, Poisson, Radon, Fourier-Borel, Laplace-Stieltjes, Z, Legendre, Mellin, Lebediv, Zak, Carson, Hodograph, Fray, Stieltjes, Gegenbauer, Convolution, Borel, Lebedev-Skal’skay, Lambert, etc.
If you want to learn more or see where I learned this from, watch Khan Academy or refer to Wolfram MathWorld.
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