On the Uniqueness of Haar Series Converging over Subsequences of Partial Sums

G. G. Gevorkyan

Yerevan State University

Abstract: We prove the existence of a sequence of positive integers $n_i$, $i\in \mathbb{N}$, that has zero density in $\mathbb{N}$ and possesses the following property: if the subsequence
$$ S_{n_i}(x)=\sum_{k=1}^{n_i}a_k\chi_k(x) $$
of partial sums of a Haar series converges everywhere to an everywhere finite integrable function $f$, then this series is the Fourier–Haar series of $f$.

Keywords: Haar system, uniqueness theorem, Fourier–Haar series.

UDC: 517.53

Received: 16.01.2025
Revised: 07.03.2025

DOI: 10.4213/mzm14621

 English version: Mathematical Notes, 2025, 118:3, 510–518

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