Certain special multiple sine series (Q2714723)

The author proves three theorems. The following one is typical.NEWLINENEWLINENEWLINETheorem 2. Let the set \({\mathcal G}\subset (0,\infty)\times (0,\infty)\) be convex with respect to both coordinate directions and contain infinitely many points with integer coordinates. Denote by \(X_{{\mathcal G}}\) the characteristic function of \({\mathcal G}\).NEWLINENEWLINENEWLINEIf the numbers \(a_{i,j}\) are decreasing to \(0\) as one of the indices is kept fixed, while the other tends to \(\infty\), NEWLINE\[NEWLINE\sum^\infty_{i=1} a_{i,i}< \infty,\tag{\(*\)}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\sum^n_{i=1} a_{i,n}+ \sum^n_{j=1} a_{n,j}\to 0\quad\text{as}\quad n\to\infty,NEWLINE\]NEWLINE then the double sine series NEWLINE\[NEWLINE\sum^\infty_{i=1} \sum^\infty_{j=1} X_{{\mathcal G}}(i,j) a_{i,j}(\sin ix)(\sin jy)\tag{\(**\)}NEWLINE\]NEWLINE converges in the sense of Pringsheim at each point \((x,y)\). Furthermore, condition \((*)\) is necessary if we require the convergence of series \((**)\) in the case of all convex sets \({\mathcal G}\).NEWLINENEWLINEFor the entire collection see [Zbl 0952.00006].