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
Hauptverfasser: Hirayama, M., Karagulyan, D.
Format: Artikel
Sprache:eng
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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.
ISSN:0133-3852
1588-273X
DOI:10.1007/s10476-024-00010-3