Equiconvergent series
Convergent or divergent series and
whose difference is a convergent series with zero sum:
. If their difference is only a convergent series, then the series are called equiconvergent in the wide sense.
If and
are functions, for example,
, where
is any set and
is the set of real numbers, then the series
and
are called uniformly equiconvergent (uniformly equiconvergent in the wide sense) on
if their difference is a series that is uniformly convergent on
with sum zero (respectively, only uniformly convergent on
).
Example. If two integrable functions on are equal on an interval
, then their Fourier series are uniformly equiconvergent on every interval
interior to
, and the conjugate Fourier series are uniformly equiconvergent in the wide sense on
.
Equiconvergent series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equiconvergent_series&oldid=18128