Uniqueness Theorems for Multiple Franklin Series

Abstract

It is proved, that if the square partial sums $\sigma_{q_n}(\textbf{x})$ 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 $f$.

A Uniqueness Theorem for Orthonormal Spline Series

“…In the multi-dimensional case, in this theorem instead of partial sums one can take (see [8]) partial sums, or (see [9]), where is any increasing sequence of natural numbers, for which the ratio ⁄ is bounded. Similar result on uniqueness is also obtained for the multi-dimensional Franklin series (see [10]).…”