ℝ²

log 12024-12-15

Properties1
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1 min read

Why didn’t anyone tell me about this theorem years ago? : r/math

In summary, what we have is

Today, I am trying to make some statement like this

(This guy solves the heat equation on with Dirichlet boundary conditions, if that’s not clear, we can ask the same (following) question for periodic/other boundary conditions also.)

For any and bounded sequence , supposedly uniformly converges to a smooth function (even analytic? 1) in both because of the factor in the coefficients.

For we will get a monstrosity like https://www.desmos.com/calculator/kywrkvgap5

So I am looking for possibilities to extend the notion of the initial

(our domain is )

  • (0) we already know in means f in
  • (1) , do we have extended Fourier series for peeps?
  • (2) bounded sequences should correspond to functions (finite essential supremum) on this could go horribly wrong lmao
  • (3) distributions, for eg what’s the “Fourier series” for Dirac delta at ?

Number of “o”s determine the difficulty level:

From


I think we understand that (1) above is at least not gonna happen.

https://mathoverflow.net/questions/28428/convergence-of-fourier-series-of-l1-functions

Footnotes

  1. https://math.stackexchange.com/a/801677/1290493 ↩

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