Geodesic
On the Fourier--Haar Coefficients of Functions of Several Variables with Bounded Vitali Variation
Matematičeskie zametki, Tome 70 (2001) no. 6, pp. 803-814.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we study the behavior of the Fourier–Haar coefficients $a_{m_1,\dots,m_n}(f)$ of functions $f$ Lebesgue integrable on the $n$-dimensional cube $D_n=[0,1]^n$ and having a bounded Vitali variation $V_{D_n}f$ on it. It is proved that $$ \sum _{m_1=2}^\infty\dotsi\sum _{m_n=2}^\infty |a_{m_1,\dots,m_n}(f)| \le\biggl(\frac{2+\sqrt 2}3\biggr)^n\cdot V_{D_n}f $$ and shown that this estimate holds for some function of bounded finite nonzero Vitali variation. 
@article{MZM_2001_70_6_a0,
     author = {S. Yu. Galkina},
     title = {On the {Fourier--Haar} {Coefficients} of {Functions} of {Several} {Variables} with {Bounded} {Vitali} {Variation}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {803--814},
     publisher = {mathdoc},
     volume = {70},
     number = {6},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2001_70_6_a0/}
}
TY  - JOUR
AU  - S. Yu. Galkina
TI  - On the Fourier--Haar Coefficients of Functions of Several Variables with Bounded Vitali Variation
JO  - Matematičeskie zametki
PY  - 2001
SP  - 803
EP  - 814
VL  - 70
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2001_70_6_a0/
LA  - ru
ID  - MZM_2001_70_6_a0
ER  - 
%0 Journal Article
%A S. Yu. Galkina
%T On the Fourier--Haar Coefficients of Functions of Several Variables with Bounded Vitali Variation
%J Matematičeskie zametki
%D 2001
%P 803-814
%V 70
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2001_70_6_a0/
%G ru
%F MZM_2001_70_6_a0
S. Yu. Galkina. On the Fourier--Haar Coefficients of Functions of Several Variables with Bounded Vitali Variation. Matematičeskie zametki, Tome 70 (2001) no. 6, pp. 803-814. http://geodesic.mathdoc.fr/item/MZM_2001_70_6_a0/

[1] Ulyanov P. L., “O ryadakh po sisteme Khaara”, Matem. sb., 63(105):3 (1964), 356–391 | MR | Zbl

[2] Galkina S. Yu., “O koeffitsientakh Fure–Khaara ot funktsii s ogranichennoi variatsiei”, Matem. zametki, 51:1 (1992), 42–54 | MR | Zbl

[3] Vitali G., “Sui gruppi di punti e sulle funzioni di variabili reale”, Atti. Accad. Sci. Torino, 43 (1908), 75–92

[4] Shvarts L., Analiz, T. 1, Mir, M., 1972

[5] Chistyakov V. V., “K teorii mnogoznachnykh otobrazhenii ogranichennoi variatsii odnoi veschestvennoi peremennoi”, Matem. sb., 189:5 (1998), 153–176 | MR | Zbl