Integrability theorems for trigonometric series

We show that, if the coefficients (a_n) in a series ${\displaystyle \frac{a_0}{2}+{\sum }_{n=1}^{\infty }{a}_n\cos \left(nt\right)}$ tend to 0 as n → ∞ and satisfy the regularity condition that ${\displaystyle {\sum }_{m=0}^{\infty }{\{{\sum }_{j=1}^{\infty }\left[{\sum }_{n=j{2}^m}^{(j+1)2^m-1}|a_n-a_{n+1}|\right]²\}}^{1/2}<\infty }$, 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 ${\displaystyle {\sum }_{n=1}^{\infty }{b}_n\sin \left(nt\right)}$ tend to 0 and satisfy the corresponding regularity condition, then the sine series represents an integrable function on [-π,π] if and only if ${\displaystyle {\sum }_{n=1}^{\infty }\frac{\left|b_n\right|}{n}<\infty }$. These conclusions were previously known to hold under stronger restrictions on the sizes of the differences ${\displaystyle \Delta a_n=a_n-a_{n+1}}$ and ${\displaystyle \Delta b_n=b_n-b_{n+1}}$. 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 ${\displaystyle a_n\log n\to 0}$ as n → ∞. The corresponding statement also holds for sine series for which ${\displaystyle {\sum }_{n=1}^{\infty }\frac{\left|b_n\right|}{n}<\infty }$. 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.