Convergence and localization in Orlicz classes for multiple Walsh-Fourier series with a lacunary sequence of rectangular partial sums

详细信息    查看全文

• 作者：S.K. Bloshanskaya ; I.L. Bloshanskii ; ^{ig.bloshn@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Lacunary sequence of rectangular partial sums ; Local smoothness condition ; Multiple Walsh&ndash ; Fourier series ; Orlicz class ; Weak generalized localization almost everywhere
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：1 March 2016
• 年：2016
• 卷：435
• 期：1
• 页码：765-782
• 全文大小：440 K

文摘

For functions f   in Orlicz classes, we consider multiple Walsh–Fourier series for which the rectangular partial sums S_n(x;f)$S_n(x;f)$ have indices n=(n₁,…,n_N)∈Z^N$n=(n_1,\dots ,n_N)\in {\mathbb{Z}}^N$ (N≥3$N\ge 3$), where either N   or N−1$N-1$ components are elements of (single) lacunary sequences. For this series, we prove the validity of weak generalized localization almost everywhere on an arbitrary measurable set A⊂I^N={x∈R^N:0≤x_j<1,j=1,2,…,N}$\mathfrak{A}\subset {\mathbb{I}}^N=\{x\in {\mathbb{R}}^N:0\le x_j<1,j=1,2,\dots ,N\}$, in the case when the structure and geometry of A$\mathfrak{A}$ are defined by the properties B_k${\mathbb{B}}_k$, 2≤k≤N$2\le k\le N$. We define the relation between the parameter k   and the “smoothness” of functions in terms of the Orlicz classes. As a consequence, we obtain some results on the “local smoothness conditions.” In particular, the theorem is proved for the convergence of Walsh–Fourier series on an arbitrary open set Ω⊂I^N$\mathrm{\Omega }\subset {\mathbb{I}}^N$ under the minimal conditions imposed on the smoothness of the function on this set.