For the orthonormal Haar system {
X n} the paper proves that for each 0 <
? < 1 there exist a measurable set
E ? 0, 1] with measure |
E | > 1 ?
? and a series of the form Σ
n=1 ∞ a n X n with
a i ↘ 0, such that for every function
f ∈
L 1(0, 1) one can find a function
\(\tilde f\) ∈
L 1(0, 1) coinciding with
f on
E, and a series of the form
$\sum\limits_{i = 1}^\infty {\delta _i a_i \chi _i } where \delta _i = 0 or 1$
, that would converge to
\(\tilde f\) in
L 1(0, 1).