Tetunashvili, Sh.; Tetunashvili, T. On the reconstruction of coefficients of Walsh series with gaps. (English) Zbl 1486.42045 Trans. A. Razmadze Math. Inst. 176, No. 1, 159-162 (2022). Formulas to calculate coefficients of a Walsh series with gaps by means of values of the sum of this series at certain two points are presented. These two points vary depending on the index of the coefficient being calculated. More precisely, the following theorem is established: let \(\{n_k\}_{k=0}^{\infty}\) be a sequence of integers such that \(2^{k}\leq n_k<2^{k+1}\) for all \(k=0, 1, \dots\) and let \(\sigma\) be any rearrangment of nonnegative integers. Let also \(\{W_n\}_{n=0}^{\infty}\) be the Walsh system and \(E\) be a Lebesgue masurable subset of \([0,1]\). Then:1.) If \(\mu E>\frac{1}{2}\), then there exists a sequence \(\{t_m\}_{m=0}^{\infty}\) of numbers from \(E\) such that if \[ \sum_{k=0}^{\infty}a_kW_{n_{\sigma(k)}}(t_{m}) = f(t_m) \] for every \(m=0, 1, \dots\) then \[ a_k=\frac{1}{2}W_{n_{\sigma(k)}}(t_{2k})[f(t_{2k})- f(t_{2k+1})] \] for every \(k= 0, 1, \dots\). 2.) If \(\mu E> 0\), then there exist an integer \(k_0\geq 0\) and a sequence \(\{t_m^\prime\}_{m=0}^{\infty}\) of numbers from \(E\) such that if \[ \sum_{k=0}^{\infty}a_kW_{n_{\sigma(k)}}(t_m^\prime) = f(t_m^\prime) \] for every \(m=0, 1, \dots\) then \[ a_k=\frac{1}{2}W_{n_{\sigma(k)}}(t_{2k}^\prime)[f(t_{2k}^\prime)- f(t_{2k+1}^\prime)] \] for every \(k=k_0, k_0+1, \dots\).This theorem generalizes corresponding statements by S. B. Stechkin and P. L. Ul’yanov [Am. Math. Soc., Transl., II. Ser. 95, 203–217 (1970; Zbl 0216.14501)] and A. V. Bakhshetsyan [Math. Notes 33, 84–90 (1983; Zbl 0517.42044); translation from Mat. Zametki 33, No. 2, 169–178 (1983)]. Reviewer: Rostom Getsadze (Uppsala) MSC: 42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 40A30 Convergence and divergence of series and sequences of functions Keywords:Walsh series; Rademacher series; rearrangement of series Citations:Zbl 0216.14501; Zbl 0517.42044 PDFBibTeX XMLCite \textit{Sh. Tetunashvili} and \textit{T. Tetunashvili}, Trans. A. Razmadze Math. Inst. 176, No. 1, 159--162 (2022; Zbl 1486.42045) Full Text: Link References: [1] A. V. Bakhshetsyan, Zeros of series in the Rademacher system. (Russian)Mat. Zametki33 (1983), no. 2, 169-178, 315. · Zbl 0517.42044 [2] B. Golubov, A. Efimov, V. Skvortsov,Walsh Series and Transforms. (Russian) Theorey and Applications, Moskow, Nauka, 1987. · Zbl 0785.42010 [3] S. B. Steˇhkin, P. L. Ul’yanov, On sets of uniqueness. (Russian)Izv. Akad. Nauk SSSR Ser. Mat.26(1962), 211-222. · Zbl 0107.05101 [4] Sh. Tetunashvili, T. Tetunashvili, On coefficients of series with respect to the Rademacher system.Proc. A. Razmadze Math. Inst.165(2014), 142-146. · Zbl 1366.40001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.