Integrability theorems for trigonometric series
Tom 107 / 1993
Streszczenie
We show that, if the coefficients (a_n) in a series $a_0/2+∑_{n=1}^∞ a_n cos(nt)$ tend to 0 as n → ∞ and satisfy the regularity condition that $∑_{m=0}^∞ {∑_{j=1}^∞ [∑_{n=j2^m}^{(j+1)2^m-1} |a_n - a_{n+1}|]²}^{1/2} < ∞$, then the cosine series represents an integrable function on the interval [-π,π]. We also show that, if the coefficients (b_n) in a series $∑_{n=1}^∞ b_n sin(nt)$ tend to 0 and satisfy the corresponding regularity condition, then the sine series represents an integrable function on [-π,π] if and only if $∑_{n=1}^∞ |b_n|/n < ∞$. These conclusions were previously known to hold under stronger restrictions on the sizes of the differences $Δa_n = a_n - a_{n+1}$ and $Δb_n = b_n - b_{n+1}$. We were led to the mixed-norm conditions that we use here by our recent discovery that the same combination of conditions implies the integrability of Walsh series with coefficients (a_n) tending to 0. We also show here that this condition on the differences implies that the cosine series converges in L¹-norm if and only if $a_n log n → 0$ as n → ∞. The corresponding statement also holds for sine series for which $∑_{n=1}^∞ |b_n|/n < ∞$. If either type of series is assumed a priori to represent an integrable function, then weaker regularity conditions suffice for the validity of this criterion for norm convergence.
