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

We determine the distance (up to a multiplicative constant) in the Zygmund class $\Lambda_{\ast}(\mathbb{R}^n)$ to the subspace $\mathrm{J}(\mathbf{bmo})(\mathbb{R}^n).$ The latter space is the image under the Bessel potential $J := (1-\Delta)^{-1/2}$ of the space $\mathbf{bmo}(\mathbb{R}^n),$ which is a non-homogeneous version of the classical $\mathrm{BMO}.$ 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 question when the Zygmund class is replaced by the H\"older space $\Lambda_{s}(\mathbb{R}^n),$ with $0 < s \leq 1$ and the corresponding subspace is $\mathrm{J}_{s}(\mathbf{bmo})(\mathbb{R}^n),$ 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 were known earlier only for $n = s = 1$ with a proof that does not extend to the general case. Our results are expressed in terms of second differences. As a byproduct of our wavelet based proof, we also obtain the distance from $f \in \Lambda_{s}(\mathbb{R}^n)$ to $\mathrm{J}_{s}(\mathbf{bmo})(\mathbb{R}^n)$ in terms of the wavelet coefficients of $f.$ We additionally establish a third way to express this distance in terms of the size of the hyperbolic gradient of the harmonic extension of $f$ on the upper half-space $\mathbb{R}^{n+1}_{+}.$