On the Uniqueness of Haar Series Converging in Subsequences of Zero Density to Denjoy Integrable Functions

A certain class of increasing sequences of positive integers ᵢ\ , i , of zero density is introduced such that, if a subsequence of partial sums Snᵢ(x)=∑ₖ₌₁nᵢaₖχₖ(x) of the Haar series converges everywhere to an everywhere finite Denjoy-integrable function f , then this series is the Fourier–Denjoy–Haar series of f . It is pointed out that, among such sequences ᵢ\ , there are some such that every interval [2ᵏ+1, 2ᵏ⁺¹] , k , contains at most one nᵢ . An example is constructed showing that, in this statement, the narrow Denjoy integral cannot be replaced by the wide Denjoy integral.