Almost everywhere convergence and divergence of Cesàro means with varying parameters of Walsh-Fourier series (Q2156697)

The authors of this paper obtain results about the almost everywhere convergence and divergence of subsequences of Cesáro means with zero tending parameters of the Walsh-Fourier series. For this aim, let us consider the interval \(\mathbb{I}=[0,1)\) and let \(x=\sum_{n=0}^\infty\frac{x_n}{2^{n+1}}\) be the dyadic expansion of \(x\in\mathbb{I}\). Given \(n\in\mathbb{N}\), they consider its binary expansion \(n=\sum_{k=0}^\infty \varepsilon_k(n)2^k\), (where \(\varepsilon_k(n)=0\) or \(\varepsilon_k(n)=1\) for \(k\in\mathbb{N}\)) and denote \(|n|=\operatorname{Max}_{j\in\mathbb{N}}\{\varepsilon_j(n)\not=0\}\). The \(n\)-th Walsh Paley function is defined as \[ w_n(x)=(-1)^{\sum_{j=0}^\infty \varepsilon_j(n)x_j}, \ \ \ x\in\mathbb{I}. \] In this context, the Walsh-Dirichlet kernel and the Fejer kernel of the Walsh-Fourier series are given respectively by \[ D_n(x)=\sum_{k=0}^{n-1}w_k(x)\ \ \ \text{and} \ \ \ K_n(x)=\frac{1}{n}\sum_{j=0}^{n-1}D_j(x). \] The partial sums of Walsh Fourier series are defined as \(S_m(f,x)=\sum_{j=0}^{m-1}\hat{f}(j)w_j(x)\), where \(\hat{f}(j)=\int_{\mathbb{I}}fw_j\) is the Walsh Fourier coefficient of the function \(f\in L^1(\mathbb{I})\). This way, the \((C,\alpha_n)\) means of the Walsh Fourier series of the function \(f\) is given by \[ \sigma_n^{\alpha_n}(f,x)=\frac{1}{A^{\alpha_n}_{n-1}}\sum_{j=1}^nA^{\alpha_n-1}_{n-j}S_j(f,x), \] where \(A^{\alpha_n}_{n-1}=\frac{(1+\alpha_n)\cdots(n+\alpha_n)}{n!}\), \(n\in\mathbb{N}\), \(\alpha_n\not=-1,-2,\dots\) For a sequence \(\alpha=\{\alpha_n\}_{n\in\mathbb{N}}\) and \(K>0\), let us consider \[ P_K(\alpha)=\left\{n\in\mathbb{N}:\frac{P(n,\alpha)}{n^{\alpha_n}}\leq K<\infty\right\}, \ \ \ \text{where} \ \ \ P(n,\alpha)=\sum_{i=0}^\infty\varepsilon_i(n)2^{i\alpha_n}. \] The authors introduce the weigthed version of the variation of an \(n\in\mathbb{N}\) by \(V(n,\alpha)=\sum_{i=0}^\infty|\varepsilon_i(n)-\varepsilon_{i+1}(n)|2^{i\alpha_n}\), with binary coefficients \(\{\varepsilon_k(n)\}_{k\in\mathbb{N}}\) and denote \(V_K(\alpha)=\left\{n\in\mathbb{N}:\frac{V(n,\alpha)}{n^{\alpha_n}}\leq K<\infty\right\}\). So, they obtain the following results, among others: \par i) Suppose \(\alpha_n\in(0,1)\). Let \(f\in L_1(\mathbb{I})\). Then, we have that \(\sigma_n^{\alpha_n}(f)\to f\), a.e., provided that \(V_k(\alpha)\ni n\to\infty\). \par ii) Let \(f\in L_1(\mathbb{I})\) and \(\lim_{n\to\infty}\frac{V(n,\alpha)}{n^{\alpha_n}}=\infty\). Then, \(\lim_{n\to\infty}\frac{n^{\alpha_n}\sigma_n^{\alpha_n}(f,x)}{V(n,\alpha)}=0,\) a.e. \par iii) Let \(f\in L_1(\mathbb{I})\). Then there exists a sequence \(\mu_j(f)\), such that for each subsequence of natural numbers with \(n_j\geq\mu_j(f)\), we have \(\sigma_{n_j}^{\alpha_{n_j}}(f)\to f\), a.e. \par iv) For each sequence of natural numbers \(v_j\nearrow\infty\), there exists a function \(f\in L_1(\mathbb{I})\) and an another sequence of natural numbers with \(N_j\geq v_j\) for which we have the everywhere divergence of \(S_{N_j}(f)\). \par v) Theorem: Let \(p>0\), then there exists a positive constant \(C_p\), such that, \(\| \operatorname{Sup}_{N\in\mathbb{N}} |f^\ast |K_{2^N}^{\alpha_N}| | \|_p \leq C_p \|f\|_{H_p}\), \(f\in H_p\), (\(H_p\) the Hardy space). \par vi) Theorem: Let \(p>0\), then there exists a positive constant \(C_p\), such that, \(\| \operatorname{Sup}_{n\in P_K(\alpha)} |f^\ast |K_{n}^{\alpha_n}| | \|_p \leq C_p\| |f| \|_{H_p}\), \(f\in H_p\).