The latest incarnation of Fourier series is in AI, in the so called RoPe technique.
The FFT is present in some of the pre-Mamba state space models as well to optimize applying the convolution.
An interactive guide to Fourier series (2022)
11–17 of 17 posts
Re: An interactive guide to Fourier series (2022)
#12I recommend learning complex numbers and the problems they can solve before touching on the main topic, Fourier series.
I considered using the complex Fourier series for the epicycles, but also wanted the article to be self-contained and concise.
I might add that in a rewrite of the article if more people feel this way.
Re: An interactive guide to Fourier series (2022)
#13"...You may be wondering - the functions and aren't periodic, how come we can still decompose them into sine/cosine sums? One trick is to set the period to infinity, and compute the series at this limit."
Wouldn't this effectively make frequency 0?
Later on for the drawings you pick period T to be equal to 1, thus frequency to be 2*pi. This does make more sense both practically and mathematically too.
P.S. A small nit, a typo "f_t and y_t" should rather be "f_x and f_y". Very fun read, thanks!
Re: An interactive guide to Fourier series (2022)
#14"...You may be wondering - the functions and aren't periodic, how come we can still decompose them into sine/cosine sums? One trick is to set the period to infinity, and compute the series at this limit." Wouldn't this effectively make frequency 0? Later on for the drawings you pick period T to be equal to 1, thus frequency to be 2*pi. This does make more sense both practically and mathematically too. P.S. A small ni…
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Re: An interactive guide to Fourier series (2022)
#15[deleted]
Re: An interactive guide to Fourier series (2022)
#16"...You may be wondering - the functions and aren't periodic, how come we can still decompose them into sine/cosine sums? One trick is to set the period to infinity, and compute the series at this limit." Wouldn't this effectively make frequency 0? Later on for the drawings you pick period T to be equal to 1, thus frequency to be 2*pi. This does make more sense both practically and mathematically too. P.S. A small ni…
> Wouldn't this effectively make frequency 0?
Yes! Letting the period length approach infinity, would make the lowest frequency (omega_0, also delta each other frequencies) approach 0, effectively turning the discrete sum of the fourier series into an integral, turning the fourier series into the continous fourier transform.
Re: An interactive guide to Fourier series (2022)
#17"...You may be wondering - the functions and aren't periodic, how come we can still decompose them into sine/cosine sums? One trick is to set the period to infinity, and compute the series at this limit." Wouldn't this effectively make frequency 0? Later on for the drawings you pick period T to be equal to 1, thus frequency to be 2*pi. This does make more sense both practically and mathematically too. P.S. A small ni…
> Wouldn't this effectively make frequency 0? Yes! Letting the period length approach infinity, would make the lowest frequency (omega_0, also delta each other frequencies) approach 0, effectively turning the discrete sum of the fourier series into an integral, turning the fourier series into the continous fourier transform.
> ...turning the fourier series into the continous fourier transform
Yes, if talking about the Fourier transform. My understanding was that author proposed that "trick" in context of discrete series for bounded non-periodic functions x(t), y(t).
Instead, what followed was more like an extension to periodic function.