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We determine the distance (up to a multiplicative constant) in the Zygmundclass $\Lambda_{\ast}(\mathbb{R}^n)$ to the subspace$\mathrm{J}(\mathbf{bmo})(\mathbb{R}^n).$ The latter space is the image underthe Bessel potential $J := (1-\Delta)^{-1/2}$ of the space$\mathbf{bmo}(\mathbb{R}^n),$ which is a non-homogeneous version of theclassical $\mathrm{BMO}.$ Locally, $\mathrm{J}(\mathbf{bmo})(\mathbb{R}^n)$consists of functions that together with their first derivatives are in$\mathbf{bmo}(\mathbb{R}^n).$ More generally, we consider the same questionwhen the Zygmund class is replaced by the H\"older space$\Lambda_{s}(\mathbb{R}^n),$ with $0 < s \leq 1$ and the corresponding subspaceis $\mathrm{J}_{s}(\mathbf{bmo})(\mathbb{R}^n),$ the image under$(1-\Delta)^{-s/2}$ of $\mathbf{bmo}(\mathbb{R}^n).$ One should note here that$\Lambda_{1}(\mathbb{R}^n) = \Lambda_{\ast}(\mathbb{R}^n).$ Such results wereknown earlier only for $n = s = 1$ with a proof that does not extend to thegeneral case.  Our results are expressed in terms of second differences. As a byproduct ofour wavelet based proof, we also obtain the distance from $f \in\Lambda_{s}(\mathbb{R}^n)$ to $\mathrm{J}_{s}(\mathbf{bmo})(\mathbb{R}^n)$ interms of the wavelet coefficients of $f.$ We additionally establish a third wayto express this distance in terms of the size of the hyperbolic gradient of theharmonic extension of $f$ on the upper half-space $\mathbb{R}^{n+1}_{+}.$

Approximation in the Zygmund and H\"older classes on $\mathbb{R}^n$