Convergence and divergence systems

Study of almost everywhere convergence of series by mean of martingale methods

“…Let (X n ) n∈N ⊂ L p (Ω, A, P) be such that E n (X n ) = 0 for every n ∈ N (p > 1). Then, for every N ∈ N, we have the following maximal inequality (13) S * N,X p ≤…”

On the convergence of ∑𝑐_{𝑛}𝑓(𝑛𝑥) and the Lip 1/2 class

“…Clearly the Fourier series of f * is absolutely convergent, and thus, by Theorem 2 of [13], (f * (n k x)) is a convergence system for any increasing sequence (n k ) of integers.…”

“…The sufficiency of (6) in Theorem 4 is contained in Theorem 2 of Gaposhkin [13]; note that for this part of the theorem we do not need any arithmetic assumptions on H. Similarly, in the case of Theorem 8, the sufficiency of (15) is contained in Theorem 7, again without any assumptions on H. To prove the converse statements, assume first that H is the set of primes, i.e. Clearly the Fourier series of f * is absolutely convergent, and thus, by Theorem 2 of [13], (f * (n k x)) is a convergence system for any increasing sequence (n k ) of integers.…”

“…Carleson's convergence theorem (see [5]) states that (f (nx)) is a convergence system in the case f (x) = sin 2πx. Using this result, Gaposhkin [13] proved the following more general criterion:…”

“…On the other hand, this condition is not necessary and sufficient: using Theorem 2 of [13] one can easily construct continuous functions f satisfying no Lipschitz condition such that (f (nx)) is a convergence system. Under some (restrictive) arithmetic conditions on the Fourier series of f , the following theorem gives a necessary and sufficient condition:…”