Abstract:
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.
Fulltext:
PDF file (210 kB)
