Uniqueness theorems for multiple Franklin series

It is proved, that if the square partial sums $\sigma_{q_n}(x)$ of a multiple Franklin series converge in measure to a function $f$, the ratio $\dfrac{q_{n+1}}{q_n}$ is bounded and the majorant of partial sums satisfies to a necessary condition, then the coefficients of the series are restored by the function $f$.