On the coexistence of convergence and divergence phenomena for integral averages and an application to the Fourier–Haar series
Let C , D ⊂ N be disjoint sets, and C = { 1 / 2 c : c ∈ C } , D = { 1 / 2 d : d ∈ D } . We consider the associate bases of dyadic, axis-parallel rectangles R C and R D . We give necessary and sufficient conditions on the sets C a n d D such that there is a positive function f ∈ L 1 ( [ 0 , 1 ) 2 ) s...
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Veröffentlicht in: | Analysis mathematica (Budapest) 2024-03, Vol.50 (1), p.149-187 |
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Hauptverfasser: | , |
Format: | Artikel |
Sprache: | eng |
Schlagworte: | |
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Zusammenfassung: | Let
C
,
D
⊂
N
be disjoint sets, and
C
=
{
1
/
2
c
:
c
∈
C
}
,
D
=
{
1
/
2
d
:
d
∈
D
}
. We consider the associate bases of dyadic, axis-parallel rectangles
R
C
and
R
D
. We give necessary and sufficient conditions on the sets
C
a
n
d
D
such that there is a positive function
f
∈
L
1
(
[
0
,
1
)
2
)
so that the integral averages are convergent with respect to
R
C
and divergent for
R
D
. We next apply our results to the two-dimensional Fourier--Haar series and characterize convergent and divergent sub-indices. The proof is based on some constructions from the theory of low-discrepancy sequences such as the van der Corput sequence and an associated tiling of the unit square. |
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ISSN: | 0133-3852 1588-273X |
DOI: | 10.1007/s10476-024-00010-3 |