Uniqueness theorems for generalized Haar systems
Let \(\{p_k\}\) be a sequence of integers \(\geq2\) and set \(m_k=p_1\cdots p_k\). The direct sum of the cyclic groups of order \(p_k\) may be realised as functions on \([0,1]\) in a way similar to the Walsh functions (corresponding to the case \(p_k=2\)). In this framework, the generalised Haar functions may be defined as the \(L^2\)-normalised functions \(\chi_n\) on \([0,1]\), where \(n=m_k+r(p_{k+1}-1)+s-1\) with \(0\leq r\leq m_k-1\) and \(1\leq s\leq p_{k+1}-1\), with support and constant modulus on the interval \([rm_k^{-1},(r+1)m_k^{-1})\), and with successive arguments \(0,2\pi sp_{k+1}^{-1},\dots,2\pi s(p_{k+1}-1)p_{k+1}^{-1}\) on its \(p_{k+1}\) subintervals of length \(m_{k+1}^{-1}\); \(\chi_0\) is defined as the constant function \(1\). Consider the (Haar-)Fourier series \(\sum a_n\chi_n\), let \(S_m=\sum_{n=0}^{m-1}a_n\chi_n\) be its partial sums, and \(S^\ast(x)=\sup|S_m(x)|\) the corresponding maximal function. The authors generalise a theorem by \textit{V. V. Kostin\/} [Math.\ Notes 73, No. 5, 662--679 (2003; Zbl 1052.42025); translation from Mat.\ Zametki 73, No. 5, 704--723 (2003)] from the case of a bounded sequence \(\{p_k\}\) to the general case and prove that if the partial sums \(S_{m_k}\) converge in measure to an a.e.\ finite measurable function \(f\) and \(\lambda_p\mathrm{meas}\{x:S^\ast(x)>\lambda_p\}=o(1)\) for some sequence \(\{\lambda_p\}\) that increases to \(\infty\), then the Fourier coefficients may be reconstructed as \(a_n=\lim_p\int_0^1[f(t)]_{\lambda_p}\overline{\chi_n(t)}dt\), where \([f(t)]_{\lambda_p}\) is the restriction of \(f\) to the set where its modulus is bounded by \(\lambda_p\). By combining their method with their results in [Proc.\ Yerevan State Univ., Phys.\ Math.\ Sci. 51, No. 1, 13--17 (2017; Zbl 1367.42011)], they provide a necessary and sufficient condition for an integrable function to admit a Fourier series as above in terms of a variant of the maximal function that takes Fejér averages on the \(p_k\) intervals \([rm_k^{-1},(r+1)m_k^{-1})\) inside intervals \([(j-1)m_{k-1}^{-1},jm_{k-1}^{-1})\).
scientific article; zbMATH DE number 6968668