The almost everywhere convergence of the dyadic Fourier series in L2 is studied. The logarithmic behaviour of the partial sums of Dyadic Fourier series in L2 is established. In order to obtain the estimation for the maximal operator corresponding to the dyadic Fourier series, the properties and asymptotical behaviour of the Dirichlet kernel are investigated. The general representation in the dyadic group and the properties of the characteristic set are used.

1.
G.
Gat
,
Acta Acad. Paed. Agriensis Sectio Matematicae
24
,
105
110
(
1997
).
2.
U.
Gyorgy Gat
and
G.
Karagulyan
,
Analysis Mathematica
40
,
243
266
(
2014
).
3.
A.
Zygmund
,
Trigonometric series
(
Cambridge University Press
,
Cambridge
,
1959
).
4.
W. F.
Schipp
,
F.
Wade
and
J.
PL
,
Walsh series: An introduction to dyadic harmonic analysis
(
Adam Hilger
,
Bristol
,
1990
).
5.
P.
Billard
,
Studia Math.
28
,
363
388
(
1967
).
6.
R. A.
Hunt
,
Actes Congres Intern. Math.
2
655
661
(
1970
).
7.
J.
Gosselin
,
Studies in Analysis, Advances in Math., Suppl. Studies
4
,
Academic Press
,
New York
,
223
232
(
1979
).
8.
Gyorgy
Gat
,
Acta Math. Hungar
116
,
209
221
(
2007
).
9.
G.
Gat
,
J. of Approx. Theory
162
,
687
708
(
2010
).
10.
N.
Memiá
,
Advance in Mathematics: Scientific Journal
4
,
65
77
(
2015
).
11.
U.
Gyorgy Gat
and
G.
Karagulyan
,
J. Math. Anal. Appl.
421
,
206
214
(
2015
).
This content is only available via PDF.