log 12024-12-15

parent:: logs

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 $u (t, x)$ solves the heat equation on $[0, 1]$ with Dirichlet boundary conditions, if that’s not clear, we can ask the same (following) question for periodic/other boundary conditions also.)

For any $t > 0$ and bounded sequence $a_{n}$ , $u$ supposedly uniformly converges to a smooth function (even analytic? ¹) in both $t, x$ because of the $e^{- n^{2}}$ factor in the coefficients.

For $a_{n} = 1$ 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

f = n \geq 1 \sum a_{n} sin (nπ x)

(our domain is $[0, 1]$ )

(0) we already know $(a_{n})$ in $l^{2}$ means f in $L^{2}$
(1) $L^{2} [0, 1] < L^{1} [0, 1]$ , do we have extended Fourier series for $L^{1}$ peeps?
(2) bounded sequences should correspond to $L^{\infty}$ functions (finite essential supremum) on $[0, 1]$ $\leftarrow$ this could go horribly wrong lmao
(3) distributions, for eg what’s the “Fourier series” for Dirac delta at $1/2$ ?