Difference and Wavelet Characterizations of Distances from Functions in Lipschitz Spaces to Their Subspaces

Let ${\mathrm{\Lambda }}_s$ denote the Lipschitz space of order $s\in (0,\mathrm{\infty })$ on ${\mathbb{R}}^n$, which consists of all $f\in \mathfrak{C}\cap L^{\mathrm{\infty }}$ such that, for some constant $L\in (0,\mathrm{\infty })$ and some integer $r\in (s,\mathrm{\infty })$, \begin{equation*} \label{0-1}Δ_r f(x,y): =\sup_{|h|\leq y} |Δ_h^r f(x)|\leq L y^s, \ x\in$\mathbb{R}$^n, \ y \in(0, 1]. \end{equation*} Here (and throughout the article) $\mathfrak{C}$ refers to continuous functions, and ${\mathrm{\Delta }}_h^r$ is the usual $r$-th order difference operator with step $h\in {\mathbb{R}}^n$. For each $f\in {\mathrm{\Lambda }}_s$ and $\epsilon \in (0,L)$, let $S(f,\epsilon ):=\{(x,y)\in {\mathbb{R}}^n\times [0,1]:\frac{{\mathrm{\Delta }}_{r}f(x,y)}{y^s}>\epsilon \}$, and let $\mu :\mathcal{B}({\mathbb{R}}_{+}^{n+1})\to [0,\mathrm{\infty }]$ be a suitably defined nonnegative extended real-valued function on the Borel $\sigma$-algebra of subsets of ${\mathbb{R}}_{+}^{n+1}$. Let $\epsilon (f)$ be the infimum of all $\epsilon \in (0,\mathrm{\infty })$ such that $\mu (S(f,\epsilon ))<\mathrm{\infty }$. The main target of this article is to characterize the distance from $f$ to a subspace $V\cap {\mathrm{\Lambda }}_s$ of ${\mathrm{\Lambda }}_s$ for various function spaces $V$ (including Sobolev, Besov–Triebel–Lizorkin, and Besov–Triebel–Lizorkin-type spaces) in terms of $\epsilon (f)$, showing that \begin{equation*} \varepsilon(f)\sim $\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}$ (f, V\cap Λ_s){Λ_s}: = \inf{g\in Λ_s\cap V} |f-g|_{Λ_s}.\end{equation*} Moreover, we present our results in a general framework based on quasi-normed lattices of function sequences $X$ and Daubechies $s$-Lipschitz $X$-based spaces.