arXiv subjects
Publications and source records attributed to Anna Kazakova.
We study the exact discretization of the $L_2$-norm in the space spanned by the first $N$ Rademacher functions. It is shown that the sufficient number of nodes for discretization with the minimal number of nodes is equal to $N$ or $N+1$ and depends on the dimension $N$. The connection with Hadamard matrices and the Hadamard conjecture is demonstrated.
This paper studies subsystems that generalize Rademacher chaos to arbitrary systems of character of compact abelian groups. The properties of $q$-lacunarity and $\frac{2d}{d+1}$-Sidon are proved for polynomial $d$-chaos constructed from $d$-dissociated subsystems of characters of compact abelian groups. Subsystems of the polynomial chaos type for unbounded Vilenkin systems are also considered and it is shown that they are not systems of $q$-lacunarity for all $q>2$ and $α$-Sidon for all $α\in [1, 2)$.
We prove the $\frac{2d}{d+1}$-Sidon inequality for a system of functions representing the most general extension of the Rademacher $d$-chaos to the $p$-ary case.
We study problems on uniqueness sets ($U$-sets) for multiple Walsh series converging over cubes and the properties of the coefficients of such series. New broad classes of $U$-sets are constructed. In particular, it is proved that hyperplanes parallel to the coordinate ones are $U$-sets. For the coefficients of multiple Walsh series converging over cubes, both the index sets on which they can be made arbitrarily large and the index sets on which these coefficients tend to zero are described.