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

摘要

The partial sum process of orthogonal expansion ∑_n≥0c_nu_n is a geometric 2-rough process, for any orthonormal system {un}n≥0 in L~2 and any sequence of numbers {c_n} satisfying ∑_n≥_0(log_2(n+1))~2|c_n|~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≥_0log_2(n+1)|c_n|~2＜∞, which is equivalent to ∫-ππ∫-ππ|f(u)-f(v)|2|sinu-v2|dudv<∞ (with f the limit function).