Let [0,T] be an intervall and let \mathbb{T}=(\tau_n)_{n\in\mathbb{N}} be a sequence of dyadic partitions of [0,T]. Let C^{\alpha}([0,T],\mathbb{R}) for 0 be a Banach space of Hölder continuous functions with norm :\|f\|_{C^{\alpha}}=\|f\|_{\infty}+|f|_{C^{\alpha}}:=\sup\limits_{t\in [0,T]}|f(t)|+\sup\limits_{\begin{matrix}s,t\in[0,T]\\ s\neq t\end{matrix}}\frac{|s-t|^{\alpha}} and \ell^{\infty}(\mathbb{R}) be the Banach space of bounded sequence with
supremum norm :\|a\|_{\infty}:=\sup\limits_{n}|a_n|. The map S\colon C^{\alpha}([0,T],\mathbb{R})\to \ell^{\infty}(\mathbb{R}) defined as :f\mapsto \left(2^{(m+1)\left(\alpha-\tfrac{1}{2}\right)}|\theta(f)^{\mathbb{T}}_{m,k}| \right)_{m,k},\qquad (m,k)\in\mathbb{N}_0\times \{0,1,\dots,2^m-1\} is an
isomorphism, where \theta(f)^{\mathbb{T}}_{m,k} are the Schauder coefficients of f along \mathbb{T} of [0,T]. The Schauder coefficients are :\theta(f)^{\mathbb{T}}_{m,k}=\langle f',h_{m,k} \rangle_{L^1}=\int_0^T f'(x)h_{m,k}(x)dx. for
Haar functions h_{m,k} based on the dyadic partition \tau_m.
Properties • The result was generalized in 2025 for general partitions. == References ==