Cesáro means with varying parameters of Walsh-Fourier series (Q6057467)

For binary expansions of \(n=\sum_{j=0}^\infty \epsilon_j(n) 2^j\) and \(x=\sum_{j=1}^\infty x_j/2^j\) where \(x\in \mathbb{I}=[0,1]\), one sets \(\rho_n(x)=(-1)^{x_n}\) and \(w_n(x)=\prod_{k=0}^\infty(\rho_{k+1}(x))^{\epsilon_k(n)}\). The Walsh-Dirichlet kernel is \(D_n(x)=\sum_{k=0}^{n-1} w_k(x)\).\par One writes \((S_Nf)(x)=\sum_{i=0}^{N-1} \widehat{f}(i) w_i(x)\) where \(\widehat{f}(i)=\langle f, w_i\rangle \) in \(L^2(\mathbb{I})\).\par The \((C,\alpha_n)\) Cesàro averages are defined for Walsh-Fourier series by \[\sigma_n^{\alpha_n}( f,x)=\frac{1}{A_{n-1}^{\alpha_n}}\sum_{j=1}^n A_{n-j}^{\alpha_n-1} (S_jf)(x)=\langle f,K_n^{\alpha_n}(\cdot\dotplus x)\rangle\] where \(A_{n}^{\alpha}=\frac{(1+\alpha)\cdots (n+\alpha)}{n!}\) when \(\alpha\) is not a negative integer and where \(x\dotplus y=\sum \vert x_j-y_j\vert /2^{j+1}\).\par For \(\alpha_n\in (0,1)\) it is known that \(A_k^{\alpha_n}\sim k^{\alpha_n}\), see \textit{T. Akhobadze} [Acta Math. Hung. 115, No. 1--2, 59--78 (2007; Zbl 1136.42004)].\par The first main result (Thm.~3.1) provides the following \(L^1\)-estimate for \(K_n^{\alpha_n}\): if \(\alpha_n\in (0,1)\) then there are \(0<c_1\leq c_2\) such that \[\frac{c_1}{n^{\alpha_n}}\sum_{k=0}^{\vert n\vert}\vert \epsilon_k(n)-\epsilon_{k+1}(n)\vert 2^{k\alpha_n}\leq \Vert K_n^{\alpha_n}\Vert_1\leq \frac{c_2}{n^{\alpha_n}}\sum_{k=0}^{\vert n\vert}\vert \epsilon_k(n)-\epsilon_{k+1}(n)\vert 2^{k\alpha_n}\, .\] This result is used to establish uniform and \(L^1\)-convergence of \((C,\alpha_n)\) means.\par For \(n\in\mathbb{N}\) one sets \(V(n,\alpha)=\frac{1}{n^{\alpha_n}}\sum_{k=1}^{\vert n\vert}\vert \epsilon_k(n)-\epsilon_{k+1}(n)\vert 2^{k\alpha_n}\) and for \(X=C_w(\mathbb{I})\) (uniformly continuous functions on \(\mathbb{I}\) with supremum norm) or \(X=L_1(\mathbb{I})\) one defines the modulus of continuity \[\omega(2^{-n}, f)_X=\sup_{h\in (0,1/2^n]}\Vert f(\cdot\dotplus h)-f(\cdot)\Vert_X .\] The following bounds are established. First (Thm.~4.1), \[\Vert \sigma_n^{\alpha_n}f -f\Vert_X\leq c_1\omega(2^{-n}, f)_X V(n,\alpha)+c_2\alpha_n \sum_{r=0}^{\vert n\vert -2} 2^{r-\vert n\vert}\omega(2^{-r}, f)_X +c_3 \omega(2^{1-\vert n\vert}, f)_X\] and (Thm.~4.2) if for some \(f\in X(\mathbb{I})\) and a subsequence \(\{m_n\}\) one has \[\omega(1/m_n, f)_X =o(\left(1/V(m_,\alpha)\right)\] then the subsequence \(\sigma_{m_n}^{\alpha_n} f\) converges in \(X\)-norm. The remaining results establish partial converses. Namely, Thm.~4.3 states that if \(\sup V(m_n,\alpha)=\infty\) then there is in turn a subsequence of \(m_n\) and a \(g\in X\) such that the moduli of continuity bound above still holds along the subsequence but \( \Vert \sigma_n^{\alpha_n}g -g\Vert_X\) does not tend to zero along the subsequence. A corresponding result for the Hardy space \(H^1\) is also established (Thm.~4.4). Here, \(\Vert f\Vert_{H^1}=\Vert E^\ast (f)\Vert_1\) where \((E_n^\ast f)(x)=\sup_{n\geq 0}( E_n\vert f\vert)(x)\) with \((E_nf)(x)=(S_{2^n}f)(x)\).