On complete characterization of coefficients of a.e. converging orthogonal series
Abstract
We characterize sequences of numbers
(
a
n
)
such that
∑
n≥1
a
n
Φ
n
converges a.e. for any orthonormal system
(
Φ
n
)
in any
L
2
-space. In our criterion, we use the set
B={
∑
m≥n
|
a
m
|
2
;n≥1}
and its information function
h
B
(t)=−
log
3
(β−α)
for
t∈(α,β]
,
[α,β]∩B={α,β}.