Whats the difference between the Theorem of Carleson and the convergence of the Fourier Series in the quadratic mean?
2026-03-30 05:40:44.1774849244
Convergence Fourier Series
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The Fourier series of any $L^{2}$ function converges in $L^{2}$ (i.e. in quadratic mean) and this is a very easy result to prove once you kow the basic facts about orthonormal functions. The very deep result of Carelson says that the Fourier series of any $L^{2}$ function converges at almost all points.