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We study the behavior of Haar coefficients in Besov and Triebel-Lizorkinspaces on $\mathbb{R}$, for a parameter range in which the Haar system is notan unconditional basis. First, we obtain a range of parameters, extending up tosmoothness $s<1$, in which the spaces $F^s_{p,q}$ and $B^s_{p,q}$ arecharacterized 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 provethat 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 itsdyadic counterpart. At the endpoint case $s=1$ and $q=\infty$, we show thatsuch 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 byBo\v{c}karev. Finally, in several endpoint cases we clarify the relationbetween dyadic and standard Besov and Triebel-Lizorkin spaces.

Haar frame characterizations of Besov-Sobolev spaces and optimal embeddings into their dyadic counterparts