Absolute convergence of the Fourier series with respect to the generalized Haar system (Q1589191)

Let \(\{p_n\}\subset\mathbb{N}\) be an increasing sequence, \(p_n\geq 2\), \(n=1,2,\dots ,\) \(m_k=p_1\dots p_k\), \(A\) a set of type \(l/{m_k}\) from \([0,1]\). If \(n\geq 2\), then \(n=m_k+r(p_{k+1}-1)+s\), where \(k=1,2,\dots;\) \(0\leq r\leq m_k-1\), \(1\leq s\leq p_{k+1}-1\). Generalized Haar functions are defined by \(\chi_1(t)\equiv 1\), \[ \chi_n(t)=\sqrt{m_k}\exp \frac{2\pi is\alpha_{k+1}(t)}{p_{k+1}} \] for \(t\in (\frac{r}{m_k},\frac{r+1}{m_k})\backslash A\) and \(\chi_n(t)=0\) in the other points and \(t=\sum_k^{\infty}\frac{\alpha_k(t)}{m_k}\), \(\alpha_k(t)=0,1,\dots,p_{k+1}-1\). As usual, \(E_n(f)_q\) be the best approximation by \(\sum_{k=1}^n b_k\chi_k\) in \(L_q[0,1]\), \(\omega(f,\delta)_q \) is the \(L_q\)- modulus of continuity, \(E_q(\lambda)=\{f\in L_q[0,1]: E_n(f)_q\leq\lambda_n\}\). Theorem 1. If \(f\in L_q[0,1]\), \(1<q<\infty\), and \(\sum_{n=1}^{\infty}n^{\frac{1}{q}-1}E_n(f)_q<\infty\), then \[ \sum_{n=1}^{\infty}|a_n(f)\chi (t)|\tag{1} \] converges uniformly on \([0,1]\) (\(a_n(f)\) -- the Fourier coefficients of \(f\)). Theorem 2. If the series (1) converges uniformly on [0,1] for all \(f\in E_q(\lambda)\), then \(\sum_{n=1}^{\infty}n^{\frac{1}{ q}-1}\lambda_n<\infty\). Similar results are obtained for \(\omega(f,\delta)_q\).