论文标题
关于非负功能的傅立叶系列的差异和收敛现象的共存
On the coexistence of divergence and convergence phenomena for the Fourier-Haar series for non-negative functions
论文作者
论文摘要
令$ \ {h_ {n，m} \} _ {n，m \ in \ mathbb {n}} $是二维haar系统，而$ s_ {n，m} f $是其傅立叶系列的对流部分总和，与某些$ f \ in l^in^in^1（0,1（0,1）^2）$ f。令$ \ mathcal {n}，\ Mathcal {M} \ subset \ mathbb {n} $为索引的两个脱节子集。我们在集合上给出了必要且充分的条件\ Mathcal {n}} s_ {n，m} f（z）= f（z）\ quad \ text {and} \ quad \ limsup_ { $$的证明使用了一些构造，从低静止序列的理论（例如范德尔的摩尔特序列和平面的相关瓷砖）中使用了一些构造。这扩展了一些早期的结果。
Let $\{H_{n,m}\}_{n,m\in \mathbb{N}}$ be the two dimensional Haar system and $S_{n,m}f$ be the rectangular partial sums of its Fourier series with respect to some $f\in L^1([0,1)^2)$. Let $\mathcal{N}, \mathcal{M}\subset \mathbb{N}$ be two disjoint subsets of indices. We give a necessary and sufficient condition on the sets $\mathcal{N}, \mathcal{M}$ so that for some $f \in L^1([0,1)^2)$, $f \geq 0$ one has for almost every $z\in [0,1)^2$ that $$ \lim_{n,m \rightarrow \infty;n,m \in \mathcal{N}}S_{n,m}f(z)=f(z)\quad \text{ and }\quad \limsup_{n,m \rightarrow \infty;n,m \in \mathcal{M}}|S_{n,m}f(z)|=\infty. $$ The proof uses some constructions from the theory of low-discrepancy sequences such as the van der Corput sequence and an associated tiling of the plane. This extends some earlier results.
