| Year | Mathematician | Achievement |
| 1807 | Fourier | First paper on the applications of Fourier analysis. |
| 1828 | Dirichlet | Convergence proof for a wide class
of functions. But not unconditionally for continuous functions. Efforts of Riemann, Weierstrass, and Dedekind over the next forty years not successful. |
| 1876 | Du Bois-Reymond | Counterexample: Continuous function whose Fourier series diverges to infinity at a point. |
| 1900 | Fejér | Proved that any continuous function may
be reconstructed from its Fourier coefficients using Cesàro summation. |
| 1926 | Kolmogorov | Lebesgue integrable function whose Fourier
series diverges everywhere. (Comment: Kolmogorov's function is not continuous, not even Riemann integrable.) |
| 1964 | Carleson | Proof that for a continuous (or even Riemann
integrable) function
the Fourier series converges almost everywhere. |