The $L\log L$ Fourier-convergence conjecture
The following describes the scope of the Lean formalization related to the following accompanying paper(s):
Scope
The $L\log L$ Fourier-convergence conjecture asks whether ordinary symmetric Fourier partial sums converge almost everywhere at this integrability scale. The formalization proves this for every measurable complex-valued function on $\mathbb T=\mathbb R/(2\pi\mathbb Z)$ satisfying $\int_{\mathbb T}|f|\log(2+|f|)<\infty$. Outside one measurable null set, $\sum_{k=-N}^{N}\widehat f(k)e^{ikx}\to f(x)$ along the full sequence $N\to\infty$.
Comparator links
| Result | Comparator statement |
|---|---|
| Almost-everywhere convergence of full symmetric Fourier partial sums | FourierLLogL.lean |