Polynomial approximation on polytopes (Q2925659)

A subset \(K\) of \(\mathbb{R}^{d}\) is said to be simple polytope if there are precisely \(d\) edges at every vertex of \(K\). If \(E_{n}(f)_{K}\) denotes the error of the best approximation of a continuous function \(f\) in \(K\) by polynomials of degree at most \(n\), it was proven in [\textit{Z. Ditzian} and \textit{V. Totik}, Moduli of smoothness. New York etc.: Springer-Verlag (1987; Zbl 0666.41001)], that you have the estimate \(E_{n}(f)_{K}\leq M (\omega_{K}^{r} (f, \frac{1}{n}) + n^{-r}\|f\|_{K})\). Here \(\omega_{K}^{r} (f, \frac{1}{n})\) denotes the \(r\)-modulus of smoothness of \(f\) that will be defined below.NEWLINENEWLINENEWLINEIn the last 25 years it remains open the problem of the validity of such a kind of estimate for non simple polytopes. Notice that the second term in the right hand side of the previous inequality is usually dominated by the first one. The aim of this monograph is to cover this gap by proving for \(d\)-dimensional convex polytopes, i.e., closed sets in \(\mathbb{R}^{d}\), with an inner point, such that they are the convex hull of finitely many points, the following Jackson-type or direct theorem:NEWLINENEWLINETheorem. Let \(K\) a \(d\)-dimensional convex polytope and \(r=1, 2, \dots\). Then, for \(n\geq rd\), \(E_{n}(f)_{K}\leq M \omega^{r}_{K} (f, \frac{1}{n})\), where \(M\) depends only on \(K\) and \(r\).NEWLINENEWLINENEWLINEThe modulus of smoothness on \(K\) is defined as follows. Let \(f\) be a continuous function on \(K\). Its \(r\)-th symmetric difference in a direction \(e,\) i.e., a vector in \(\mathbb{R}^{d}\) with \(\|e\|=1\), is \(\Delta^{r}_{he}f(x)= \sum _{k=0}^{r}(-1)^{k} \frac{r(r-1) \cdots (r-k+1)}{k!} f(x+ (\frac{r}{2}-k) he)\) assuming that this is \(0\) if \(x + \frac{r}{2} he\) or \(x- \frac{r}{2} he\) does not belong to \(K\). Then \(\omega_{K}^{r} (f, \delta)= \sup_{\|e\|=1} \sup_{h\leq \delta} \|\Delta^{r}_{h\tilde{d}_{K}(e, x) e} f(x)\|_{K}.\) Here \(\tilde{d}_{K}(e,x)\) is the so-called normalized distance from \(x\) to the boundary of \(K\) in the direction of \(e.\) Notice that the modulus of smoothness is just the supremum of all moduli of smoothness on chords of \(K\).NEWLINENEWLINENEWLINEThe matching weak converse of the above theorem (known in the literature as Stechkin-type or converse theorem) reads as NEWLINE\[NEWLINE\omega^{r}_{K} (f, \frac{1}{n}) \leq M n^{-r} \sum_{k=0}^{n} (k+1) ^{r-1} E_{k}(f)_{K}.NEWLINE\]NEWLINE As a consequence of these results, you get that if \(f\) can be approximated with error \(n^{-\alpha}\) on any chord \(I\) of \(K\) by polynomials (of a single variable on \(I\)) of degree at most \(n=1,2, \dots\), then \(E_{n}(f)_{K}\leq M n^{-\alpha},\) where \(M\) depends only on \(K\) and \(\alpha\).NEWLINENEWLINENEWLINEThe proof of the theorem follows several steps. First, in Chapter 6 an estimate of the moduli of smoothness is deduced on small parts of a given pyramid \(S\) in \(\mathbb{R}^{3}\) satisfying the following properties: (i) no two base edges of \(S\) are parallel and (ii) the height of \(S\) lies in the interior of \(S\), up to its two endpoints. Second, in Chapter 7 a local approximation on the sets \(K_{a}= S_{a} \backslash S_{a/4}\) where \(S_{a}=S \cap\{(x_{1}, x_{2}, x_{3}) , 0 \leq x_{1} \leq a \}\) is obtained. Third, a global approximation of \(f\) on \(S_{1/64}\) is deduced in Chapter 9. Thus, you get the completion of the proof when \(d=3\) and you can extend it to the general dimension based on the following two facts (Chapter 11). NEWLINE{\parindent=6mm \begin{itemize}\item[1.] You cut off around every vertex \(V_{j}\) of \(K\) a \(d\)-dimensional pyramid \(S_{j}\) with properties (i) and (ii). \item[2.] If \(S\subset\mathbb{R}^{d}\) is a \(d\)-dimensional pyramid with apex at the origin and with base lying in the hyperplane \(\{(x_{1}, \dots, x_{d}, x_{1}=2\}\) such that (i) and (ii) hold for \(S\), then for any \(f\in C(S)\) and for any \(n\geq rd\) there is a polynomial \(p_{n}(x_{1},\dots,x_{d})\) of degree at most m such that \(\|F_{n}- p_{n}\|_{S_{n}^{*}} \prec \omega^{r}_{S} (f, \frac{1}{n}),\) where \(F_{n}(x)= f (x- v_{n})\), \(v_{n}= (- L/{n^{2}}, 0, \dots, 0)\) and \(S_{n}^{*}= S \cap \{x, \frac{2^{9r+1}}{n^{2}} \leq x_{1}\leq \frac{1}{16}\}.\) NEWLINENEWLINE\end{itemize}} Finally, for a convex polytope in \(\mathbb{R}^{d},\) consider the \(K\)-functional \(\mathcal{K}_{r} (f,t)= \inf_{g} (\|f-g\|_{K} + t \sup_{\|e\|=1} \|\tilde{d}^{r}_{K}(e,\cdot) \frac{\partial^{r}g}{\partial e^{r}}\|_{K})\), where the infimum is considered for all \(g\in C^{r}(K)\). Then, there is a constant \(M,\) depending only on \(r\) and \(K,\) such that for every \(f\in C(K)\) and for every \(0<\delta\leq 1\) you have \( \frac{1}{M} \mathcal{K}_{r} (f,\delta^{r})\leq \omega^{r}_{K} (f, \delta)\leq M \mathcal{K}_{r} (f,\delta^{r}),\) i.e., the equivalence of the moduli of smoothness and the above \(K\)-functional. This appears in Chapter 12. Notice that this result does not follow from the main theorem.NEWLINENEWLINENEWLINEThe second part of the monograph (Chapters 13--21) deals with the best approximation problem in \(L^{p}\)-spaces, \(1\leq p <\infty.\) For sake of simplicity, the author focus the attention in the case \(d=3\). Consider the \(L^{p}\) modulus of smoothness \(\omega^{r}_{K} (f, \delta)_{p}= \sup_{\|e\|=1} \sup_{h\leq \delta} (\int_{K} | \Delta^{r}_{h\tilde{d}_{K} (e,x)e} f(x)|^{p} dx) ^{1/p}.\) Thus, the main theorem reads now as follows.NEWLINENEWLINENEWLINETheorem. Let \(K\subset\mathbb{R}^{3}\) be a \(3\)-dimensional convex polytope and \(r\geq 1\). Then for \(n\geq 3r\) and \(f\in L^{p}(K),\) \(E_{n}(f)_{L^{p}(K)} \leq M \omega^{r}_{K} (f, \frac{1}{n})_{p}\), where \(M\) depends only on \(K\), \(r\), and \(p\).NEWLINENEWLINEThe Stechkin-type result is also proved. Indeed, \(\omega^{r}_{K} (f, \frac{1}{n})_{p}\leq M n^{-r} \sum_{k=0}^{n} (k+1)^{r-1} E_{k}(f)_{L^{p}(K)}\) is deduced along standard lines as in Theorem 12.2. (12.2.4) in the above mentioned monograph by Ditzian and Totik.NEWLINENEWLINEAs a consequence, the \(n^{-\alpha}\) rate of approximation along chords in a given direction implies for global approximation the rate \(n^{-\alpha}.\)NEWLINENEWLINEA stronger version of the above theorem is presented. It reads as follows:NEWLINENEWLINETheorem. Let \(K\subset\mathbb{R}^{3}\) be a \(3\)-dimensional convex polytope and \(r\geq 1\). Then, there exists a finite set \(\mathcal{E}^{*}\) of directions that depends only on \(K\), such that for \(n\geq 3r\) and \(f\in L^{p}(K),\) \(E_{n}(f)_{L^{p}(K)} \leq M \omega^{r}_{K,\mathcal{E}^{*} } (f, \frac{1}{n})_{p}\), where \(M\) depends only on \(K\), \(r\), and \(p\).NEWLINENEWLINEHere \(\omega^{r}_{K,\mathcal{E}^{*}}\) is the restriction of the modulus of smoothness to the set of directions \(\mathcal{E}^{*}\).NEWLINENEWLINENEWLINENotice that there is an extensive literature on polynomial approximation in several variables, where various moduli of smoothness are constructed for special sets like balls and spheres (in particular, \textit{Y. Xu} [Constr. Approx. 21, No. 1, 1--28 (2005; Zbl 1069.33014)], as well as \textit{F. Dai} and \textit{Y. Xu} [Adv. Math. 224, No. 4, 1233--1310 (2010; Zbl 1193.41010)]) which solve the problem there. In many cases the moduli of smoothness are equivalent to a \(K\)-functional and thus the approximation follows using such a \(K\)-functional. These do not work on polytopes and the absence of the relevant \(K\)-functional is the reason of the present approach.NEWLINENEWLINENEWLINEDespite the technicalities, the presentation of this monograph reads in a friendly way and the structure in short chapters allows the reader to follow and understand the different steps on the proofs of the main results.