Nonlinear Piecewise Polynomial Approximation and Multivariate $BV$ spaces of a Wiener--L.~Young Type. I

The named space denoted by $V_{pq}^k$ consists of $L_q$ functions on $[0,1)^d$ of bounded $p$-variation of order $k\in\mathbb N$. It generalizes the classical spaces $V_p(0,1)$ ($=V_{p\infty}^1$) and $BV([0,1)^d)$ ($V_{1q}^1$ where $q:=\frac d{d-1}$) and closely relates to several important smoothness spaces, e.g., to Sobolev spaces over $L_p$, $BV$ and $BMO$ and to Besov spaces. The main approximation result concerns the space $V_{pq}^k$ of \textit{smoothness} $s:=d\left(\frac1p-\frac1q\right)\in(0,k]$. It asserts the following: Let $f\in V_{pq}^k$ are of smoothness $s\in(0,k]$ and $N\in\mathbb N$. There exist a family $\Delta_N$ of $N$ dyadic subcubes of $[0,1)^d$ and a piecewise polynomial $g_N$ over $\Delta_N$ of degree $k-1$ such that $$\|f-g_N\|_q\leqslant CN^{-s/d}|f|_{V_{pq}^k}.$$ This implies the similar results for the above mentioned smoothness spaces, in particular, solves the going back to the 1967 Birman--Solomyak paper \cite{BS} problem of approximation of functions from $W_p^k([0,1)^d)$ in $L_q([0,1)^d)$ when ever $\frac kd=\frac1p-\frac1q$ and $q<\infty$.