Thanks again for the well explained article! The best insights for me is explaining why is it natural to define the inner product of function that way. However, is it possible to elaborate on why having discovered that cos(x) and sin(x) are orthogonal to each other helps the discovery of Fourier transformation?
However, I don't talk about Fourier transformation here. Regarding that, the main "discovery" (at least in hindsight) was that it turns convolution into the pointwise product, thus it is essentially an isometry between certain function algebras.
Thanks again for the well explained article! The best insights for me is explaining why is it natural to define the inner product of function that way. However, is it possible to elaborate on why having discovered that cos(x) and sin(x) are orthogonal to each other helps the discovery of Fourier transformation?
Thanks! In the context of Fourier series, I have written a follow-up article: https://thepalindrome.substack.com/p/the-unreasonable-effectiveness-of
However, I don't talk about Fourier transformation here. Regarding that, the main "discovery" (at least in hindsight) was that it turns convolution into the pointwise product, thus it is essentially an isometry between certain function algebras.