We study the behavior of Haar coefficients in Besov and Triebel-Lizorkin spaces on $\mathbb{R}$, for a parameter range in which the Haar system is not an unconditional basis. First, we obtain a range of parameters, extending up to smoothness $s<1$, in which the spaces $F^s_{p,q}$ and $B^s_{p,q}$ are characterized in terms of doubly oversampled Haar coefficients (Haar frames). Secondly, in the case that $1/p<s<1$ and $f\in B^s_{p,q}$, we actually prove that the usual Haar coefficient norm, $\|\{2^j\langle f, h_{j,\mu}\rangle\}_{j,\mu}\|_{b^s_{p,q}}$ remains equivalent to $\|f\|_{B^s_{p,q}}$, i.e., the classical Besov space is a closed subset of its dyadic counterpart. At the endpoint case $s=1$ and $q=\infty$, we show that such an expression gives an equivalent norm for the Sobolev space $W^{1}_p(\mathbb{R})$, $1<p<\infty$, which is related to a classical result by Bo\v{c}karev. Finally, in several endpoint cases we clarify the relation between dyadic and standard Besov and Triebel-Lizorkin spaces.

翻译：本文研究实数轴$\mathbb{R}$上Besov空间和Triebel-Lizorkin空间中哈尔系数的性质，参数范围选取在哈尔系统不构成无条件基的情形。首先，我们获得一个参数范围（光滑度$s<1$的情形下可扩展），在此范围内空间$F^s_{p,q}$和$B^s_{p,q}$可通过双重过采样哈尔系数（哈尔帧）进行刻画。其次，当$1/p<s<1$且$f\in B^s_{p,q}$时，我们实际上证明了通常的哈尔系数范数$\|\{2^j\langle f, h_{j,\mu}\rangle\}_{j,\mu}\|_{b^s_{p,q}}$仍等价于$\|f\|_{B^s_{p,q}}$，即经典Besov空间是其二元对应空间的闭子集。在端点情形$s=1$且$q=\infty$时，我们证明该表达式给出了Sobolev空间$W^{1}_p(\mathbb{R})$（$1<p<\infty$）的等价范数，这关联于Bočkarev的经典结果。最后，在若干端点情形中，我们阐明二元Besov空间和Triebel-Lizorkin空间与标准空间之间的关系。