In the following we assume that
\(\varOmega \subset {\mathbb {R}} ^{s}\) is a convex body containing
\(\sigma \) and that the sampled function
\( f:\varOmega \) \(\rightarrow {\mathbb {R}} \) is of class
\(C^{d,1}\left( \varOmega \right) \), that is
\(f\in C^{d}\left( \varOmega \right) \) and all its partial derivatives of order
d$$\begin{aligned} D^{\alpha }f=\left( \prod \limits _{i=1}^{s}\frac{\partial ^{\alpha _{i}}}{\partial \xi _{i}^{\alpha _{i}}}\right) f=\frac{\partial ^{\left| \alpha \right| }f}{\partial \xi _{1}^{\alpha _{1}}\partial \xi _{2}^{\alpha _{2}}\dots \partial \xi _{s}^{\alpha _{s}}},\text { }\left| \alpha \right| =d, \end{aligned}$$
(8)
are Lipschitz continuous in
\(\varOmega \). Let
\(K\subseteq \varOmega \) compact convex: we equip the space
\(C^{d,1}\left( K \right) \) with the semi-norm [
13]
$$\begin{aligned} \left\| f\right\| _{d,1}^{K}=\sup \left\{ \frac{\left| D^{\alpha }f({\mathbf {u}})-D^{\alpha }f({\mathbf {v}})\right| }{\left\| {\mathbf {u}}- {\mathbf {v}}\right\| _{2}}:{\mathbf {u}},{\mathbf {v}}\in K ,\text { }{\mathbf {u}} \ne {\mathbf {v}},\text { }\left| \alpha \right| =d\right\} \end{aligned}$$
(9)
and we denote by
\(T_{d}\left[ f,\overline{{\mathbf {x}}}\right] \left( \mathbf {x }\right) \) the truncated Taylor expansion of
f of order
d centered at
\( \overline{{\mathbf {x}}}\in \varOmega \)$$\begin{aligned} T_{d}\left[ f,\overline{{\mathbf {x}}}\right] \left( {\mathbf {x}}\right) =\sum _{l=0}^{d}\frac{D_{{\mathbf {x}}-\overline{{\mathbf {x}}}}^{l}f\left( \overline{{\mathbf {x}}}\right) }{l!} \end{aligned}$$
(10)
and by
\(R_{T}\left[ f,\overline{{\mathbf {x}}}\right] \left( {\mathbf {x}}\right) \) the corresponding remainder term in integral form [
19]
$$\begin{aligned} R_{T}\left[ f,\overline{{\mathbf {x}}}\right] \left( {\mathbf {x}}\right) =\int _{0}^{1}\frac{D_{\mathbf {x-}\overline{{\mathbf {x}}}}^{d+1}f\left( \overline{{\mathbf {x}}}+t\left( \mathbf {x-}\overline{{\mathbf {x}}}\right) \right) }{d!}\left( 1-t\right) ^{d}dt, \end{aligned}$$
(11)
where [
6, Ch. 4]
$$\begin{aligned} D_{\mathbf {x-}\overline{{\mathbf {x}}}}^{l}f\left( \mathbf {\cdot }\right) :=\left( \left[ D^{\beta }f\left( \mathbf {\cdot }\right) \right] _{\left| \beta \right| =1}\cdot \left( \mathbf {x-}\overline{\mathbf {x }}\right) \right) ^{l}=\sum \limits _{\left| \beta \right| =l}\frac{l! }{\beta !}D^{\beta }f\left( \mathbf {\cdot }\right) \left( \mathbf {x-} \overline{{\mathbf {x}}}\right) ^{\beta },\text { }l\in {\mathbb {N}}, \end{aligned}$$
(12)
with the multi-indices
\(\beta \) following the order specified in Sect.
1.Let us denote by
\(\ell _{i}\left( {\mathbf {x}}\right) \) the
\(i^{th}\) bivariate fundamental Lagrange polynomial. Since
$$\begin{aligned} \ell _{i}\left( {\mathbf {x}}_{j}\right) =\delta _{ij}=\left\{ \begin{array}{ll} 1, &{} \quad i=j, \\ 0, &{} \quad \text {otherwise,} \end{array} \right. \end{aligned}$$
by setting for each
\(i=1,\dots ,m,\)$$\begin{aligned} \begin{array}{c} \delta ^{i}=\left[ \begin{array}{lllllll} 0 &{} \dots &{} 0 &{} 1 &{} 0 &{} \dots &{} 0 \end{array} \right] ^{T} \\ \quad \quad \overset{\uparrow }{i^{th}~\text {column}} \end{array} \end{aligned}$$
and by solving the linear system
\(V_{\overline{{\mathbf {x}}}}\left( \sigma \right) {\mathbf {a}}^{i}=\delta ^{i}\), we get the expression of
\(\ell _{i}\left( {\mathbf {x}}\right) \) in the translated canonical basis (
3), that is
$$\begin{aligned} \ell _{i}\left( {\mathbf {x}}\right) =\sum _{\left| \alpha \right| \le d}a_{\alpha }^{i}\left( {\mathbf {x}}-\overline{{\mathbf {x}}}\right) ^{\alpha }, \end{aligned}$$
(13)
where
\({\mathbf {a}}^{i}=\left[ a_{\alpha }^{i}\right] _{\left| \alpha \right| \le d}^{T}\).
It is worth noting that the analysis developed in [
14] in connection with the estimation of the error of Hermite interpolation in
\({\mathbb {R}}^{s}\) can be used to obtain analogous bounds with respect to those obtained in (
31) and (
32). To do this, in line with [
14], given an integer
\(l\in {\mathbb {N}}\), we denote with
\(D^{l}f\left( {\mathbf {x}}\right) \) the
l-th derivative of
f in a point
\( {\mathbf {x}}\in \varOmega \), that is the
l linear operator
$$\begin{aligned} \begin{array}{cccc} D^{l}f\left( {\mathbf {x}}\right) :&\underset{l\text { times}}{\underbrace{ {\mathbb {R}} ^{s}\times \dots \times {\mathbb {R}} ^{s}}}\longrightarrow & {} {\mathbb {R}} \end{array} \end{aligned}$$
which acts on the canonical basis element
\(\underset{\nu {_{1}\text { times}}}{(\underbrace{{\mathbf {e}}_{1},\dots ,{\mathbf {e}}_{1}}},\underset{\nu _{2}\text { times}}{\underbrace{{\mathbf {e}}_{2},\dots ,{\mathbf {e}}_{2}}},\dots ,\underset{ \nu {_{s}\text { times}}}{\underbrace{{\mathbf {e}}_{s},\dots ,{\mathbf {e}}_{s}}})\) ,
\(\nu =\left( \nu _{1},\nu _{2},\dots ,\nu _{s}\right) \in {\mathbb {N}} _{0}^{s}\),
\(\left| \nu \right| =l\), as follows
$$\begin{aligned} D^{l}f\left( {\mathbf {x}}\right) \cdot \underset{\nu {_{1}\text { times}}}{( \underbrace{{\mathbf {e}}_{1},\dots ,{\mathbf {e}}_{1}}},\underset{\nu _{2}\text { times}}{\underbrace{{\mathbf {e}}_{2},\dots ,{\mathbf {e}}_{2}}},\dots ,\underset{ \nu {_{s}\text { times}}}{\underbrace{{\mathbf {e}}_{s},\dots ,{\mathbf {e}}_{s}}} )=D^{\nu }f\left( {\mathbf {x}}\right) , \end{aligned}$$
(36)
where we use the previously introduced notations for partial derivatives (
8). As an
l linear operator, the norm of
\( D^{l}f\left( {\mathbf {x}}\right) \) is defined in a standard way as follows
$$\begin{aligned} \left\| D^{l}f\left( {\mathbf {x}}\right) \right\| =\sup \limits _{\begin{array}{c} \left\| {\mathbf {u}}_{i}\right\| _{2}=1 \\ {\mathbf {u}} _{i}\in {\mathbb {R}} ^{s},1\le i\le l \end{array}}\left| D^{l}f\left( {\mathbf {x}}\right) \cdot \left( {\mathbf {u}}_{1},\dots ,{\mathbf {u}}_{l}\right) \right| , \end{aligned}$$
(37)
and therefore, for each
\(\nu \in {\mathbb {N}}_{0}^{s}\),
\(\left| \nu \right| =l\), we have
$$\begin{aligned} \left| D^{\nu }f\left( {\mathbf {x}}\right) \right| \le \left\| D^{l}f\left( {\mathbf {x}}\right) \right\| . \end{aligned}$$
(38)
By introducing the Sobolev semi-norm
$$\begin{aligned} \left| f\right| _{l,p,\varOmega }=\left( \int _{\varOmega }\left\| D^{l}f\left( {\mathbf {x}}\right) \right\| ^{p}d{\mathbf {x}}\right) ^{\frac{1}{ p}}, \end{aligned}$$
which is meaningful for functions
\(f\in W^{d+1,p}\left( \varOmega \right) \) for each
\(d+1\ge l\), from [
14, Theorem 2.1] by taking
\( p=+\infty \), we get using (
38)
$$\begin{aligned} \left| D^{\nu }\left( f-p\left[ {\mathbf {y}},\sigma \right] \right) \left( {\mathbf {x}}\right) \right|\le & {} \left\| D^{l}\left( f-p\left[ {\mathbf {y}},\sigma \right] \right) \left( {\mathbf {x}}\right) \right\| \\\le & {} \left| f-p\left[ {\mathbf {y}},\sigma \right] \right| _{l,\infty ,K} \\\le & {} \frac{1}{\left( d+1\right) !}\left( \sum \limits _{i=1}^{m}\left| \ell _{i}\right| _{l,\infty ,K}\right) \left| f\right| _{d+1,\infty ,K}\left( 2h\right) ^{d+1}, \end{aligned}$$
for any
\(\overline{{\mathbf {x}}}\in \varOmega ,\) \(\mathbf {x\in }K\mathbf {=} B_{h}\left( \overline{{\mathbf {x}}}\right) \cap \varOmega \) and for any multi-index
\(\nu \) of length
\(\left| \nu \right| =l\le d\). In order to point out links between proof of Proposition 3 and that one given in [
14, Theorem 2.1], we start with the inequality
$$\begin{aligned} \left\| D^{l}\left( f-p\left[ {\mathbf {y}},\sigma \right] \right) \left( {\mathbf {x}}\right) \right\| \le \sum \limits _{i=1}^{m}\left| R_{T}[f, {\mathbf {x}}_{i}]\left( {\mathbf {x}}\right) \right| \left\| D^{l}\ell _{i}\left( {\mathbf {x}}\right) \right\| , \end{aligned}$$
(39)
which is already stated in the paper [
14, Page 414, line 9] for the general case of Hermite interpolation. By the linearity of the operator
\(D^{l}\) with respect to the function argument, we get by using the expression (
16) of Lagrange polynomials
$$\begin{aligned} D^{l}\ell _{i}\left( {\mathbf {x}}\right) =\sum \limits _{\left| \alpha \right| \le d}a_{\alpha ,h}^{i}h^{-\left| \alpha \right| }D^{l}\left( {\mathbf {x}}-\overline{{\mathbf {x}}}\right) ^{\alpha }. \end{aligned}$$
(40)
We can explicitly compute the expression of
\(D^{l}\left( {\mathbf {x}}- \overline{{\mathbf {x}}}\right) ^{\alpha }\) when applied to a vector
\(\left( {\mathbf {u}}_{1},\dots ,{\mathbf {u}}_{l}\right) \in \left( {\mathbb {R}} ^{s}\right) ^{l}\). In order to simplify the notations, in line with [
14, Section 2], we assume that
$$\begin{aligned} \lambda =\left( \lambda _{1},\lambda _{2},\dots ,\lambda _{l}\right) \in \left\{ 1,\dots ,s\right\} ^{l}, \end{aligned}$$
and, denoting with
\(\left\{ \varepsilon _{1},\varepsilon _{2},\dots ,\varepsilon _{s}\right\} \) the canonical basis of
\( {\mathbb {R}} ^{s}\), we set
$$\begin{aligned} \varepsilon _{\lambda ,l}=\varepsilon _{\lambda _{1}}+\dots +\varepsilon _{\lambda _{l}}, \end{aligned}$$
and
$$\begin{aligned} \varepsilon _{\lambda }^{l}=\left( \varepsilon _{\lambda _{1}},\dots ,\varepsilon _{\lambda _{l}}\right) \in \left( {\mathbb {R}} ^{s}\right) ^{l}. \end{aligned}$$
Therefore
$$\begin{aligned} D^{l}\left( {\mathbf {x}}-\overline{{\mathbf {x}}}\right) ^{\alpha }\cdot \left( {\mathbf {u}}_{1},\dots ,{\mathbf {u}}_{l}\right) =\sum \limits _{\begin{array}{c} \left| \varepsilon _{\lambda ,l}\right| =l \\ \varepsilon _{\lambda ,l}\le \alpha \end{array}}\frac{\alpha !}{\left( \alpha -\varepsilon _{\lambda ,l}\right) !}\left( {\mathbf {x}}-\overline{{\mathbf {x}}}\right) ^{\alpha - \varepsilon _{\lambda ,l}}\left( {\mathbf {u}}_{1},\dots ,{\mathbf {u}}_{l}\right) ^{ \varepsilon _{\lambda }^{l}}. \end{aligned}$$
(41)
Using (
40) and (
41), we get
$$\begin{aligned} \left\| D^{l}\ell _{i}\left( {\mathbf {x}}\right) \right\|= & {} \sup \limits _{\begin{array}{c} \left\| {\mathbf {u}}_{i}\right\| _{2}=1 \\ {\mathbf {u}} _{i}\in {\mathbb {R}}^{s},1\le i\le l \end{array}}\left| D^{l}\ell _{i}\left( {\mathbf {x}}\right) \cdot \left( {\mathbf {u}}_{1},\dots ,{\mathbf {u}}_{l}\right) \right| \nonumber \\= & {} \sup \limits _{\begin{array}{c} \left\| {\mathbf {u}}_{i}\right\| _{2}=1 \\ {\mathbf {u}}_{i}\in {\mathbb {R}}^{s},1\le i\le l \end{array}}\left| \sum \limits _{\left| \alpha \right| \le d}a_{\alpha ,h}^{i}h^{-\left| \alpha \right| }D^{l}\left( {\mathbf {x}}-\overline{ {\mathbf {x}}}\right) ^{\alpha }\cdot \left( {\mathbf {u}}_{1},\dots ,{\mathbf {u}} _{l}\right) \right| \nonumber \\= & {} \sup \limits _{\begin{array}{c} \left\| {\mathbf {u}}_{i}\right\| _{2}=1 \\ {\mathbf {u}}_{i}\in {\mathbb {R}}^{s},1\le i\le l \end{array}}\left| \sum \limits _{\left| \alpha \right| \le d}a_{\alpha ,h}^{i}h^{-\left| \alpha \right| }\sum \limits _{\begin{array}{c} \left| \varepsilon _{\lambda ,l}\right| =l \\ \varepsilon _{\lambda ,l}\le \alpha \end{array}}\frac{\alpha !}{\left( \alpha -\varepsilon _{\lambda ,l}\right) !}\left( {\mathbf {x}}-\overline{{\mathbf {x}}}\right) ^{\alpha -\varepsilon _{\lambda ,l}}\left( {\mathbf {u}}_{1},\dots ,{\mathbf {u}}_{l}\right) ^{\varepsilon _{\lambda }^{l}}\right| . \nonumber \\ \end{aligned}$$
(42)
From the bound (
39), the equality (
42) and by recalling that [
13, Lemma 2.1]
$$\begin{aligned} \left| R_{T}[f,{\mathbf {x}}_{i}]\left( {\mathbf {x}}\right) \right| \le \frac{s^{d}}{\left( d-1\right) !}\left\| {\mathbf {x}}-{\mathbf {x}} _{i}\right\| _{2}^{d+1}\left\| f\right\| _{d,1}^{K}, \end{aligned}$$
we obtain
$$\begin{aligned}&\left| D^{\nu }\left( f-p\left[ {\mathbf {y}},\sigma \right] \right) \left( {\mathbf {x}}\right) \right| \nonumber \\&\quad \le \left\| D^{l}\left( f-p\left[ {\mathbf {y}},\sigma \right] \right) \left( {\mathbf {x}}\right) \right\| \nonumber \\&\quad \le \left\| f\right\| _{d,1}^{K}\frac{s^{d}}{\left( d-1\right) !} \sum \limits _{i=1}^{m}\left\| {\mathbf {x}}-{\mathbf {x}}_{i}\right\| _{2}^{d+1} \nonumber \\&\qquad \times \sup \limits _{\begin{array}{c} \left\| {\mathbf {u}}_{i}\right\| _{2}=1 \\ {\mathbf {u}}_{i}\in {\mathbb {R}} ^{s},1\le i\le l \end{array}}\left| \sum \limits _{\left| \alpha \right| \le d}a_{\alpha ,h}^{i}h^{-\left| \alpha \right| }\sum \limits _{\begin{array}{c} \left| \varepsilon _{\lambda ,l}\right| =l \\ \varepsilon _{\lambda ,l}\le \alpha \end{array}}\frac{\alpha !}{\left( \alpha -\varepsilon _{\lambda ,l}\right) !}\left( {\mathbf {x}}-\overline{{\mathbf {x}}}\right) ^{\alpha -\varepsilon _{\lambda ,l}}\left( {\mathbf {u}}_{1},\dots ,{\mathbf {u}} _{l}\right) ^{\varepsilon _{\lambda }^{l}}\right| , \nonumber \\ \end{aligned}$$
(43)
which is a slight different version of the bound (
31) given in Proposition
3. In order to have an analogous version of the bound (
32), we evaluate at
\( \overline{{\mathbf {x}}}\) the estimation (
43), and consequently we get
$$\begin{aligned} \left| D^{\nu }\left( f-p\left[ {\mathbf {y}},\sigma \right] \right) \left( \overline{{\mathbf {x}}}\right) \right|\le & {} \left\| f\right\| _{d,1}^{K}\frac{s^{d}}{\left( d-1\right) !}\sum \limits _{i=1}^{m}\left\| \overline{{\mathbf {x}}}-{\mathbf {x}}_{i}\right\| _{2}^{d+1} \nonumber \\&\times \sup \limits _{\begin{array}{c} \left\| {\mathbf {u}}_{i}\right\| _{2}=1 \\ {\mathbf {u}}_{i}\in {\mathbb {R}} ^{s},1\le i\le l \end{array}}\left| \sum \limits _{\left| \alpha \right| \le d}a_{\alpha ,h}^{i}h^{-\left| \alpha \right| }\sum \limits _{\begin{array}{c} \left| \varepsilon _{\lambda ,l}\right| =l \\ \varepsilon _{\lambda ,l}=\alpha \end{array}}\frac{\alpha !}{\left( \alpha -\varepsilon _{\lambda ,l}\right) !}\left( {\mathbf {u}}_{1},\dots ,{\mathbf {u}}_{l}\right) ^{\varepsilon _{\lambda }^{l}}\right| .\nonumber \\ \end{aligned}$$
(44)
We notice that the
s-tuple
\(\varepsilon _{\lambda ,l}\) in the internal sum depends on
\(\alpha \), but its length is equal to
\(l\le d\) and therefore we get
$$\begin{aligned} \left| D^{\nu }\left( f-p\left[ {\mathbf {y}},\sigma \right] \right) \left( \overline{{\mathbf {x}}}\right) \right|\le & {} \left\| f\right\| _{d,1}^{K}\frac{s^{d}}{\left( d-1\right) !}\sum \limits _{i=1}^{m}\left\| \overline{{\mathbf {x}}}-{\mathbf {x}}_{i}\right\| _{2}^{d+1} \nonumber \\&\times \sup \limits _{\begin{array}{c} \left\| {\mathbf {u}}_{i}\right\| _{2}=1 \\ {\mathbf {u}}_{i}\in {\mathbb {R}} ^{s},1\le i\le l \end{array}}\left| \sum \limits _{\left| \alpha \right| =l}a_{\alpha ,h}^{i}h^{-\left| \alpha \right| }\alpha !\left( {\mathbf {u}}_{1},\dots ,{\mathbf {u}}_{l}\right) ^{\varepsilon _{\lambda }^{l}}\right| \nonumber \\\le & {} \left\| f\right\| _{d,1}^{K}\frac{s^{d}}{\left( d-1\right) !} h^{d+1} \nonumber \\&\times \sum \limits _{i=1}^{m}\left| \sum \limits _{\left| \alpha \right| =l}a_{\alpha ,h}^{i}h^{-\left| \alpha \right| }\alpha !\right| \sup \limits _{\begin{array}{c} \left\| {\mathbf {u}}_{i}\right\| _{2}=1 \\ {\mathbf {u}}_{i}\in {\mathbb {R}} ^{s},1\le i\le l \end{array}}\left| \left( {\mathbf {u}}_{1},\dots ,{\mathbf {u}} _{l}\right) ^{\varepsilon _{\lambda }^{l}}\right| \nonumber \\\le & {} \left\| f\right\| _{d,1}^{K}\frac{s^{d}}{\left( d-1\right) !} h^{d-l+1}\sum \limits _{i=1}^{m}\left| \sum \limits _{\left| \alpha \right| =l}\alpha !a_{\alpha ,h}^{i}\right| . \end{aligned}$$
(45)
It is easy to see that, despite the bounds (
32) and (
45) are similar, their comparison depends on the sign of the coefficients
\(a_{\alpha ,h}^{i}\),
\(\left| \alpha \right| =l\), since the sum
\(\sum \limits _{\left| \alpha \right| =l}\alpha !a_{\alpha ,h}^{i}\) contains the term
\(\nu !a_{\nu ,h}^{i}\).