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High-Order Approximation of Set-Valued Functions

Abstract

We introduce the notion of metric divided differences of set-valued functions. With this notion we obtain bounds on the error in set-valued metric polynomial interpolation. These error bounds lead to high-order approximations of set-valued functions by metric piecewise-polynomial interpolants of high degree. Moreover, we derive high-order approximation of set-valued functions by local metric approximation operators reproducing high-degree polynomials.

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Correspondence to Alona Mokhov.

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Communicated by Wolfgang Dahmen.

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Appendices

Appendix A: Proof of Lemma 5.12

For the readers’ convenience we recall here the notation and Lemma 5.12. Let

$$\begin{aligned} W=\{(x,y):\; a'\le x \le b',\, c \le y \le d \, \}, \quad \; a\le a' < b' \le b, \quad x^*\in (a',b'),\end{aligned}$$

and let \({g_i(x): [a', x^*]\rightarrow (c,d)}\), \(i=1,2\) be such that \(g_1(x^*)=g_2(x^*)\) and \(g_1(x)<g_2(x)\) for \(x \in [a', x^*)\). Assume that

$$\begin{aligned} \mathrm {Graph}(F) \bigcap W = W \setminus \left\{ (x,y) : \; x \in [a',x^*]\, , \, y \in \big ( g_1(x),g_2(x) \big ) \right\} . \end{aligned}$$
(13)

Note that by (13) the only boundaries of \(\mathrm {Graph}(F)\) in the interior of W are \(g_1(x)\) and \(g_2(x)\). In the left neighborhood of \(x^*\), \(F(x)\bigcap [c,d]\) consists of two disjoint intervals, while in the right neighborhood of \(x^*\), \(F(x)\bigcap [c,d] = [c,d]\) (see Fig. 3).

Lemma 5.12.

Under the above notation and condition (13), assume that F has bounded second divided differences in a neighborhood of \(x^*\).

  1. (i)

    Then the function \({g(x)=g_2(x)-g_1(x)}\), \(x \in [a', x^*]\) has a vanishing left derivative at \(x=x^*\).

  2. (ii)

    If in addition \(g_i(x)\), \(i=1,2\) are continuous, then for \(\tilde{x} \in (a',x^* -\varepsilon )\), with \(\varepsilon >0\) small enough, the second divided differences of \(g_i(x)\), \(i=1,2\) in a neighborhood of \(\tilde{x}\) are bounded.

Proof

See Fig. 5 for illustration of the two claims.

(i) Let \(x_1=x^*-h\), \(x_2=x^*+h\), \(\tilde{x}=x^*\) and \(\chi _2=\{x_1, x_2\}\), \({{\widetilde{\chi }}_2=\{\tilde{x}, x_1, x_2\}}\). Denote \(y^*=\frac{g_1(x_1)+g_2(x_1)}{2}\). In accordance with Definition 5.1 we define \(f \in \Upsilon ({\widetilde{\chi }}_2,F)\). Let \({f(x_1)=g_2(x_1)}\), \({f(x_2)=y^*}\), \(f(x^*)=y^*\). It is easy to see that

$$\begin{aligned} \big ( f(x_1),f(x_2) \big ) \in \Pi ({F(x_1)},{F(x_2)}). \end{aligned}$$

Denote by \(p_1(\chi _2,f)(x)\) the linear polynomial interpolating the data \((x_1, f(x_1))\), \((x_2, f(x_2))\), then

$$\begin{aligned} (f(x^*), p_1(\chi _2,f)(x^*)) \in \Pi ({F(x^*)},{P_1(\chi _2,F)(x^*)}), \end{aligned}$$

since \(p_1(\chi _2,f)(x^*)) \) is a projection of \(f(x^*)=y^*\) on \(P_1(\chi _2,F)(x^*)\). Thus \(f[x^*,x_1,x_2] \in F[x^*;x_1,x_2].\)

Next we compute \(f[x^*,x_1,x_2]\),

$$\begin{aligned} \begin{aligned} f[x^*,x_1,x_2]&=\left( \frac{f(x_2)-f(x^*)}{h} - \frac{f(x^*)-f(x_1)}{h} \right) \frac{1}{2h}\\&=\left( \frac{y^*-y^*}{h} - \frac{y^*-g_2(x_1)}{h} \right) \frac{1}{2h} \\&= \frac{g_1(x_1)-g_2(x_1)}{4h^2} = -\frac{g(x_1)}{4h^2} = \frac{g(x^*)-g(x_1)}{h} \frac{1}{4h} \in F[x^*;x_1,x_2]. \end{aligned} \end{aligned}$$
(14)

By the assumptions and by (14) we have \(\left| \frac{g(x_1)}{4h^2} \right| =\left| \frac{g(x^*-h)}{4h^2} \right| \le |F[x^*;x_1,x_2]|\le M\) for h small enough and some \(M \in {\mathbb {R}}\).

Therefore \({ \lim _{h \rightarrow 0} \left| h\right| \left| \frac{g(x^*-h)}{h^2} \right| =0 }\). Since \(g(x^*)=0\) we get

$$\begin{aligned} \lim _{h \rightarrow 0} \left| h\right| \left| \frac{g(x^*)-g(x^*-h)}{h^2}\right| = \lim _{h \rightarrow 0} \left| \frac{g(x^*)-g(x^*-h)}{h} \right| =g'_{-}(x^*)=0. \end{aligned}$$
Fig. 5
figure 5

Graph of F in violet , graph of \(P_1(\chi _2,F)\) restricted to \([x_1,x_2]\) in brown (Color figure online)

(ii) For \(\tilde{x} \in (a',x^*)\) let \(g(\tilde{x})=L>0\) and \(x_1=\tilde{x}-h_1\), \(x_2=\tilde{x}+h_2\), with \(h_i>0\), \(x_i \in (a', x^*)\), \(i=1,2\). Let \(h=\max \{h_1, h_2 \}\) and \(\chi _2=\{x_1, x_2\}\). It is enough to show that \(g_i[\tilde{x}, x_1, x_2] \in F[\tilde{x}; x_1, x_2]\), \(i=1,2\). We prove it for \(i=1\). The proof for \(i=2\) is similar.

According to Definition 5.1, we have to show that for h small enough

$$\begin{aligned} \big (g_1(x_1), g_1(x_2)\big ) \in \Pi ({F(x_1)},{F(x_2)}), \end{aligned}$$
(15)

and

$$\begin{aligned} \big ( g_1(\tilde{x}), p_1(\chi _2, g_1)(\tilde{x}) \big ) \in \Pi ({F(\tilde{x})},{ P_1(\chi _2, F)(\tilde{x})}). \end{aligned}$$
(16)

First we prove (16). Denote \(I=[a',x^*-\varepsilon ]\). By the continuity of \(g_i(x)\), \(i=1,2\) we have for h small enough

$$\begin{aligned} \left| g_i(x_j)-g_i(\tilde{x})\right| \le \omega _{I}\big ( {g_i},{h} \big ) < \frac{L}{4},\; i,j=1,2. \end{aligned}$$
(17)

Now, using the last inequality and the inverse triangle inequality we get

$$\begin{aligned} \left| g_2(x_2)-g_1(x_1)\right|&= \left| g_2(x_2)-g_2(\tilde{x}) + g_2(\tilde{x})-g_1(\tilde{x}) + g_1(\tilde{x}) -g_1(x_1) \right| \\&\ge -\left| g_2(x_2)-g_2(\tilde{x})\right| + \left| g_2(\tilde{x})-g_1(\tilde{x})\right| - \left| g_1(\tilde{x}) -g_1(x_1)\right| \\&\ge L-\omega _{I}\big ( {g_1},{h} \big )-\omega _{I}\big ( {g_2},{h} \big ) > \frac{L}{2}. \end{aligned}$$

Denote \(\lambda =\frac{h_2}{h_1+h_2}\), \(y_i=p_1(\chi _2,g_i)(\tilde{x})\), \(i=1,2\). By Lemma 3.2 and by (17) we get

$$\begin{aligned} \left| g_1(\tilde{x})-y_1\right| \le \omega _{I}\big ( {g_1},{h} \big ) < \frac{L}{4}. \end{aligned}$$
(18)

On the other hand

$$\begin{aligned} \begin{aligned} \left| g_1(\tilde{x})-y_2\right|&= \left| g_1(\tilde{x}) - g_2(\tilde{x}) +g_2(\tilde{x}) - y_2\right| \\&\ge \left| g_1(\tilde{x})-g_2(\tilde{x})\right| - \lambda \left| g_2(\tilde{x})- g_2(x_1)\right| - (1-\lambda )\left| g_2(\tilde{x})-g_2(x_2))\right| \\&\ge L - \omega _{I}\big ( {g_2},{h} \big ) > \frac{3L}{4}, \end{aligned}\nonumber \\ \end{aligned}$$
(19)

and similarly

$$\begin{aligned} \left| g_2(\tilde{x})-y_1\right| \ge L - \omega _{I}\big ( {g_1},{h} \big ) > \frac{3L}{4}. \end{aligned}$$
(20)

Now, consider four cases:  (i) \(y_1 < g_1(\tilde{x})\), (ii) \(y_1=g_1(\tilde{x})\), (iii) \(g_1(\tilde{x})< y_1 < g_2(\tilde{x})\) and (iv) \( y_1 \ge g_2(\tilde{x})\).

The second case gives (16) immediately, while the fourth case \( y_1 \ge g_2(\tilde{x})\) is impossible for h small enough by (18) and (20).

In the first case \(g_1(\tilde{x}) \notin P_1(\chi _2,F)(\tilde{x})\). Therefore \(\Pi _{P_1(\chi _2,F)(\tilde{x})}{(g_1(\tilde{x}))} \in \partial P_1(\chi _2,F)(\tilde{x}) = \{y_1,y_2\}\). By (18) and (19) for h small enough (16) follows.

In the third case \(y_1 \notin F(\tilde{x})\). Therefore \(\Pi _{F(\tilde{x})}{(y_1)} \in \partial F(\tilde{x}) =\{g_1(\tilde{x}),g_2(\tilde{x})\}\). Again in view of (18) and (19) we conclude (16) for h small enough.

The proof of (15) is based on similar arguments as the proof of (16). \(\square \)

Appendix B: Verification of Example 5.13

We recall the definition of F. Let \(g_1(x), g_2(x): [a,b] \longrightarrow [0, \infty )\) such that \(-g_1<g_2\), with \(g_i''\) continuous, non-positive and \(g_i' \ge 0\) in [ab], \(i=1,2\). Also let \({\tau (x): [a,c] \longrightarrow [0, \infty )}\), \(a<c<b\) with \(\tau ' <0, \) \(\tau ''\) positive and continuous in [ac]. Moreover \({\tau ^{(i)}(c)=0}\), \(i=0,1,2\) and \({0< \tau (a) < \min \{ g_2(a), g_1(a) \} }\). The multifunction F is defined by

$$\begin{aligned} F(x) = \left\{ \begin{array}{ll} { [-g_1(x)\, ,-\tau (x)]\, \bigcup \, [\, \tau (x), g_2(x) \,] }\, , &{} x \in [a,c], \\ {[-g_1(x)\, , g_2(x) ]} \, , &{} x \in [c,b]. \\ \end{array} \right. \end{aligned}$$

To analyze F we first consider a simpler SVF with a convex graph

$$\begin{aligned} G(x) = [ -g_1(x)\, , g_2(x) ] \; , \; \, x \in [a,b]. \end{aligned}$$

The graphs of F and G are presented in Fig.6.

Fig. 6
figure 6

Graph of G (left) and graph of F (right)

For \(\chi =\{x_1, x_3 \}\) such that \(a \le x_1 < x_3 \le b\) and \(x_2 \in (x_1, x_3)\) we show that

$$\begin{aligned} \big | \, G[x_2; x_1, x_3] \, \big | \le M = \frac{1}{2} \max \left\{ \, |g''_i(x)|\; , \; x\in (x_1,x_3) , i=1,2 \, \right\} . \end{aligned}$$

First, we consider all metric pairs of the sets \(G(x_1), G(x_3)\),

$$\begin{aligned} \Pi ({G(x_1)},{G(x_3)}) =&\{(y,y)\, : \, y \in [ -g_1(x_1)\, , g_2(x_1) ] \} \\&\bigcup _{j=1}^2 \left\{ \, (\, (-1)^j g_j(x_1), y\, )\,: \; y \in \mathrm {co}\{(-1)^j g_j(x_1), (-1)^j g_j(x_3)\} \right\} . \end{aligned}$$

Thus \(P_1(\chi , G)\) consists of lines , l(x) , restricted to \([x_1,x_3]\)

$$\begin{aligned} \begin{aligned}&P_1(\chi , G)(x) = \{ \, l(x)\equiv y \, : \,y \in [-g_1(x_1), g_2(x_1)] \, \} \\&\bigcup _{j=1}^2 \left\{ \, l(x)=(-1)^j g_j(x_1) +\frac{y-(-1)^j g_j(x_1)}{x_3-x_1}(x-x_1)\,: \right. \\&\quad \left. \; y \in \mathrm {co}\{(-1)^j g_j(x_1), (-1)^j g_j(x_3)\} \right\} . \end{aligned} \end{aligned}$$
(21)

Since \(g''_1\) and \(g''_2\) are negative, \(\mathrm {Graph}\left( G|_{[x_1,x_3]}\right) \) is convex, therefore

$$\begin{aligned} \mathrm {Graph}\left( P_1(\chi , G)|_{[x_1,x_3]} \right) \subset \mathrm {Graph}\left( G|_{[x_1,x_3]} \right) . \end{aligned}$$

Thus it follows that \(\left( l(x_2),l(x_2) \right) \in \Pi ({P_1(\chi , G)(x_2)},{G(x_2)})\). By this and the definition of the metric divided difference all the triplets \(\left( l(x_1), l(x_2), l(x_3) \right) \) of the lines in (21) contribute zero to \(G[x_2; x_1,x_3]\).

It remains to consider the rest of the metric pairs in \(\Pi ({P_1(\chi , G)(x_2)},{G(x_2)})\). Let \(l_j\) denote the line through \(\big (x_1, (-1)^jg_j(x_1) \big )\) and \(\big (x_3, (-1)^jg_j(x_3) \big )\), \(j=1,2\), restricted to \([x_1, x_3]\). The lines \(l_1\) and \(l_2\) are the lower and the upper boundaries of \(\mathrm {Graph}\left( P_1(\chi , G)|_{[x_1,x_3]} \right) \), respectively. It is clear that

$$\begin{aligned} \big \{ (l_j(x_2), y )\, : \; y \in \mathrm {co}\{ l_j(x_2), (-1)^j g_j(x_2) \} \subset \Pi ({P_1(\chi , G)(x_2)},{G(x_2)}) \big \}, \; j=1,2 \ , \end{aligned}$$

and the corresponding second-order divided differences are

$$\begin{aligned} \begin{aligned}&\bigcup _{j=1}^2 \left\{ \, f[x_2, x_1, x_3]\, : \, f(x_i)=(-1)^j g_j(x_i), i=1,3 \, , \,\right. \\&\quad \left. f(x_2)=y\in \mathrm {co}\{ l_j(x_2) , (-1)^j g_j(x_2) \} \right\} . \end{aligned} \end{aligned}$$
(22)

First, note that the divided differences in (22) for \(j=2\) have absolute values bounded by the constant \(M_2\), where \({M_j = \frac{1}{2} \max \{ |g_j''(x)|\, : \, x\in [a,b]\} }\), \(j=1,2\). This is obvious for \(y=g_2(x_2)\), since the divided difference equals \(g_2[x_1, x_2, x_3]\) , which by the assumption on \(g_2\), is non-positive with absolute value bounded by \(M_2\). Note that all the other divided differences in (22) for \(j=2\) are in the interval \(\big [ g_2[x_1, x_2, x_3] , 0 \big ]\). Indeed, in view of the relation (obtained from (7) for \(r=3\)) we have

$$\begin{aligned} f[x_2, x_1, x_3] = \frac{f(x_1)}{(x_1-x_2)(x_1-x_3)} +\frac{f(x_2)}{(x_2-x_1)(x_2-x_3)} + \frac{f(x_3)}{(x_3-x_1)(x_3-x_2)} \, , \end{aligned}$$

and since \({f(x_2)=y \in [l_2(x_2),g_2(x_2) ]}\) and \({f(x_i)=g_2(x_i)=l_2(x_i)}\), \(i=1,3\), while the denominator of \(f(x_2)\) is negative, we get \(f[x_2, x_1, x_3] \in \big [g_2[x_1, x_2, x_3] , 0 \big ]\).

For the case \(j=1\) in (22), by similar arguments one can show that all the divided differences are in \({ \big [0, -g_1[x_1, x_2, x_3] \big ] \subset [0, M_1]}\).

In the next step we prove that \(\{ |F[x_2; x_1, x_3]|\, : \, a\le x_1<x_2<x_3\le b \}\) is a bounded set, with a bound independent of \(x_1, x_2, x_3\). Let for \(j=1,2\)

$$\begin{aligned} h_j(x)= \left\{ \begin{array}{ll} (-1)^j\tau (x)\, , &{} x \in [a,c], \\ 0 \, , &{} x \in [c,b]. \\ \end{array} \right. \end{aligned}$$

By the assumptions on \(\tau (x)\), we have \({h_1''(x) \le 0}\), \({h_2''(x) \ge 0}\) for \({x\in [a,b]}\) and that \({h_1'(x)>0}\), \({h_2'(x)<0}\), for \(x \in [a,c]\). Note that we can decompose F(x) as \(F(x)=F_1(x)\bigcup F_2(x)\) with \({F_1(x)=[-g_1(x), h_1(x)]}\) and \({F_2(x)=[h_2(x), g_2(x)]}\). The multifunctions \(F_1, F_2\) satisfy \(F_1(x)\bigcap F_2(x)=\emptyset \) for \(x \in [a,c)\) and \({F_1(x)\bigcap F_2(x)=\{0\}}\) for \(x \in [c,b]\). Also the boundaries of \(F_j\), \(j=1,2\) satisfy all the requirements that the boundaries of G satisfy. Then from the result on the second order divided differences of G we conclude that \(\{ |F_j[x_2; x_1, x_3]| \} \le {\widetilde{M}}_j\) for any \(a\le x_1<x_2<x_3\le b\) with \({ {\widetilde{M}}_j = \frac{1}{2} \max \left\{ \, |g''_j(x)| , |h_j''(x)|\, : \; x\in [a,b] \right\} }\), \(j=1,2\).

To obtain the boundedness of divided differences of F it remains to show that for any \({a\le x_1<x_2<x_3\le b}\)

$$\begin{aligned} F[x_2; x_1, x_3] \subseteq F_1[x_2; x_1, x_3] \bigcup F_2[x_2; x_1, x_3]\, . \end{aligned}$$
(23)

Indeed, it is easy to conclude (23) from the following observation

$$\begin{aligned} \Pi ({F(x_1)},{F(x_2)}) \subseteq \Pi ({F_1(x_1)},{F_1(x_2)}) \bigcup \Pi ({F_2(x_1)},{F_2(x_2)}) \end{aligned}$$

and from the convexity of the graphs of \(F_1\) and \(F_2\), the monotonicity of their boundaries, as well as the fact that \(h_2(x)=-h_1(x)\), \(x \in [a,b]\).

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Dyn, N., Farkhi, E. & Mokhov, A. High-Order Approximation of Set-Valued Functions. Constr Approx (2022). https://doi.org/10.1007/s00365-022-09572-7

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Keywords

  • Set-valued functions
  • Metric linear combinations
  • Set-valued metric divided differences
  • Set-valued metric polynomial interpolation
  • Metric local linear operators
  • High-order approximation

Mathematics Subject Classification

  • 26E25
  • 41A10
  • 41A25
  • 41A35
  • 41A36
  • 41H04