The partial sum process of orthogonal expansions as geometric rough process with Fourier series as an example—An improvement of Menshov–Rademacher theorem

The partial sum process of orthogonal expansion ∑n⩾0cnun∑n⩾0cnun is a geometric 2-rough process, for any orthonormal system {un}n⩾0{un}n⩾0 in L2L2 and any sequence of numbers {cn}{cn} satisfying ∑n⩾0(log2(n+1))2|cn|2<∞∑n⩾0(log2(n+1))2|cn|2<∞. Since being a geometric 2-rough process implies the existence of a limit function up to a null set, our theorem could be treated as an improvement of Menshov–Rademacher theorem. For Fourier series, the condition can be strengthened to ∑n⩾0log2(n+1)|cn|2<∞∑n⩾0log2(n+1)|cn|2<∞, which is equivalent to ∫−ππ∫−ππ|f(u)−f(v)|2|sinu−v2|dudv<∞ (with f the limit function).