Walsh-Fourier series and their generalizations in Orlicz spaces
Let \(m= \{m_j, j= 0,1,2,\dots\}\) be a sequence of integers such that \(2\leq m_j\leq C\), for some positive number \(C\), \(j= 0,1,2,3,\dots\), and let \(G_m\) denote the group of integer sequences of the form \[ x= (x_0, x_1,x_2,\dots, x_k,\dots),\quad 0\leq x_k\leq m_k, \] with group operation \(x+ y= (x_0\dot+ y_0,\dots, x_k\dot+ y_k,\dots)\), \(x_k\dot+ y_k= (x_k+ y_k)\text{ mod }m_k\). If \(M_{k+ 1}= m_0m_1\cdots m_k\), \(M_0= 1\), and \((n_0, n_1,\dots, n_k,\dots)\), is the sequence in terms of which an integer \(n\) is representable as a series involving \(M_k\), \(k= 0,1,2,3,\dots\), let \(\phi_n(x)= \prod_{k= 0} r_k(x)^{n_k}\), where \(r_k(x)= \exp(2\pi ix_k/m_k)\), \(x\in G_m\). Then the functions \(\phi_n\), \(n= 0,1,2,\dots\), constitute the Walsh-Vilenkin complete orthonormal system. If \(D_n= \sum^{n- 1}_{j= 0}\phi_j\), \(n= 1,2,3,\dots\), are the Dirichlet kernels, then the Fourier series of \(f\) in \(L^1(G_m)\) has partial sums \(S^\phi_n(f)\), representable as \(S^\phi_n(f)(x)= \int_{G_m} f(t) D_n(x- t)d\mu(t)\), where \(\mu\) is the normalized Haar measure on \(G_m\). In the main results of this paper, the authors show that if \(N\), \(Q\) are increasing nonnegative functions on \((0,\infty)\), \(N\) satisfies a \(\Delta_2\)-condition and \(Q\) is convex, and if \(R_N(u)= u\int^u_1 t^{-2}N(t)dt\), then \(R_N(t)\leq CQ(t)\) for \(t> t_0> 0\), for constants \(C\), \(t_0\), is equivalent to \[ \int_{G_m} N(| S^\phi_n(f)|) d\mu\leq C\Biggl(1+ \int_{G_m} Q(| f|\Biggr)d\mu. \] Statements in the paper provide references of previously derived \(L^p\)-estimates.
Walsh-Vilenkin complete orthonormal system
scientific article; zbMATH DE number 1217019
scientific article