Post by Baron von Lotsov on Nov 16, 2023 4:27:29 GMT
I've got a bit of interesting maths here which is both highly abstract and has practical uses at the same time.
First of all we start off with the concept of a Hilbert space, formally defined as a "Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space." When we talk about a space it is important to understand space can be any number of dimensions, so a straight line is a one dimensional space, a flat plane is a 2 dimensional space, and we live in a three dimensional space, although you can go right up to an infinite number of dimensions. It turns out that a Fourier transform is an example of a Hilbert space. Say we had a sound with a particular waveform which we plot as amplitude vs time, then as we take the Fourier transform of this function we get a plot which is amplitude vs frequency. You often see this if you hit a musical note and get the fundamental frequency and a series of harmonics at multiples of the frequency. We could view these different components of the sound as different dimensions in a multi-dimensional space, one dimension for each harmonic. In the general case of a sound we can expect frequencies all along the frequency axis, so this is how we get an infinite dimensional space. You also have to have a function which eventually dies out as you go to infinity, e.g. a Gaussian would do this.
Anyway, the Fourier transform is totally general and will transform any function from a spacial coordinate to a frequency coordinate system and in the following example we are dealing with heat. To make matters simple we only consider one dimension of the heat, like heat transfer long a metal bar. At time = 0 we have a certain distribution of heat vs the x position and we want to know what this looks like after a particular time. We can set up a partial differential equation to describe the transfer of heat. However to solve this there is a neat trick involving transposing this into the Fourier transform and using the dot product of functions in Hilbert space. There is a bit of mental contortion to get your head around to see how this works.
I've got two videos here to explain it. The first one is how a dot product of two functions works as a prerequisite to understanding how we solve this heat problem as explained in the second. Both videos are very nicely explained too.