Equiconvergent series

Convergent or divergent series $\sum_{n=1}^\infty a_n$ and $\sum_{n=1}^\infty b_n$ whose difference is a convergent series with zero sum: $\sum_{n=1}^\infty(a_n-b_n)=0$. If their difference is only a convergent series, then the series are called equiconvergent in the wide sense.

If $a_n=a_n(x)$ and $b_n=b_n(x)$ are functions, for example, $a_n,b_n\colon X\to\mathbf R$, where $X$ is any set and $\mathbf R$ is the set of real numbers, then the series $\sum_{n=1}^\infty a_n(x)$ and $\sum_{n=1}^\infty b_n(x)$ are called uniformly equiconvergent (uniformly equiconvergent in the wide sense) on $X$ if their difference is a series that is uniformly convergent on $X$ with sum zero (respectively, only uniformly convergent on $X$).

Example. If two integrable functions on $[-\pi,\pi]$ are equal on an interval $I\subset[-\pi,\pi]$, then their Fourier series are uniformly equiconvergent on every interval $I^*$ interior to $I$, and the conjugate Fourier series are uniformly equiconvergent in the wide sense on $I^*$.