On one estimate of divided differences and its applications

TL;DR

This paper provides an estimate for divided differences with coalescing points, strengthening classical inequalities in Hermite interpolation and deriving new approximation bounds for smooth functions.

Contribution

It introduces a novel estimate for divided differences with coalescing points and applies it to improve Whitney and Marchaud inequalities in Hermite interpolation.

Findings

Strengthened bounds for Hermite interpolation errors.

Derived approximation estimates involving moduli of smoothness.

Improved understanding of divided differences with multiple coalescing points.

Abstract

We give an estimate of the general divided differences $[x_0,\dots,x_m;f]$, where some of the $x_i$'s are allowed to coalesce (in which case, $f$ is assumed to be sufficiently smooth). This estimate is then applied to significantly strengthen Whitney and Marchaud celebrated inequalities in relation to Hermite interpolation. For example, one of the numerous corollaries of this estimate is the fact that, given a function $f\in C^{(r)}(I)$ and a set $Z=\{z_j\}_{j=0}^\mu$ such that $z_{j+1}-z_j \geq \lambda |I|$, for all $0\le j \le \mu-1$, where $I:=[z_0, z_\mu]$, $|I|$ is the length of $I$ and $\lambda$ is some positive number, the Hermite polynomial ${\mathcal L}(\cdot;f;Z)$ of degree $\le r\mu+\mu+r$ satisfying ${\mathcal L}^{(j)}(z_\nu; f;Z) = f^{(j)}(z_\nu)$, for all $0\le \nu \le \mu$ and $0\le j\le r$, approximates $f$ so that, for all $x\in I$, \[ \big|f(x)- {\mathcal L}(x;f;Z) \big| \le C \left( \mathop{\rm dist}\nolimits(x, Z) \right)^{r+1} \int_{\mathop{\rm dist}\nolimits(x, Z)}^{2|I|}\frac{\omega_{m-r}(f^{(r)},t,I)}{t^2}dt , \] where $m :=(r+1)(\mu+1)$, $C=C(m, \lambda)$ and $\mathop{\rm dist}\nolimits(x, Z) := \min_{0\le j \le \mu} |x-z_j|$.

Equations

$$\big{|}f(x)-{\mathcal{L}}(x;f;Z)\big{|}\leq C\left(\mathop{\rm dist}\nolimits(x,Z)\right)^{r+1}\int_{\mathop{\rm dist}\nolimits(x,Z)}^{2|I|}\frac{{\omega}_{m-r}(f^{(r)},t,I)}{t^{2}}dt,$$

$$|[x_0,x_1,x_2;f]|\le \frac{c}{x_2-x_0}{\int }_h^{x_2-x_0}\frac{{\omega }_2(f,t)}{t^2}dt,$$

$${\Delta }_u^k(f,x;[a,b]):=\{\begin{array}{ll}{\sum }_{i=0}^k(-1{)}^i(\genfrac{}{}{0ex}{}{k}{i})f(x+(k/2-i)u),& x\pm (k/2)u\in [a,b],\\ 0,& \mathrm{otherwise},\end{array}$$

$${\omega }_k(f,t;[a,b]):=\underset{0<u\le t}{\sup }\Vert {\Delta }_u^k(f,\cdot ;[a,b]){\Vert }_{C[a,b]}$$

$$[x_0;f]:=f(x_0),$$

$$[x_0,\dots ,x_m;f]=[\underset{m+1}{\underbrace{x_0,\dots ,x_0}};f]:=\frac{f^{(m)}(x_0)}{m!}.$$

$$[x_0,\dots ,x_m;f]:=\frac{1}{x_{j^{\ast }}-x_0}\left([x_1,\dots ,x_m;f]-[x_0,\dots ,x_{j^{\ast }-1},x_{j^{\ast }+1},\dots ,x_m;f]\right),$$

$$L_m(x;f)$$

$$:=f(x_0)+\sum _{j=1}^m[x_0,\dots ,x_j;f](x-x_0)\dots (x-x_{j-1})$$

$$L_m^{(l_j-1)}(x_j;f)=f^{(l_j-1)}(x_j),\text{for all }0\le j\le m.$$

$$[x_0,\dots ,x_m;f]:=\frac{f(x_{j_{\ast }})-L_{m-1}(x_{j_{\ast }};f;x_0,\dots ,x_{j_{\ast }-1},x_{j_{\ast }+1},\dots ,x_m)}{{\prod }_{j=0,j\ne j_{\ast }}^m(x_{j_{\ast }}-x_j)}.$$

$$x_j<x_{j+r+1},\text{for all}0\le j\le m-r-1.$$

$${\mathcal{Q}}_{m,r}:=$$

$$=$$

$$d(p,q):=d(p,q;X):=\min \{x_{q+1}-x_p,x_q-x_{p-1}\},$$

$$d:=d(X):=d(0,m;X)=2(x_m-x_0).$$

$${\Lambda }_{p,q,r}(x_0,\dots ,x_m;\phi ):=\frac{{\int }_{x_q-x_p}^{d(p,q)}{u}^{p+r-q-1}\phi (u)du}{\prod _{i=0}^{p-1}(x_q-x_i)\prod _{i=q+1}^m(x_i-x_p)},(p,q)\in {\mathcal{Q}}_{m,r},$$

$${\Lambda }_r(x_0,\dots ,x_m;\phi ):=\underset{(p,q)\in {\mathcal{Q}}_{m,r}}{\max }{\Lambda }_{p,q,r}(x_0,\dots ,x_m;\phi ).$$

$$\left|[x_0,\dots ,x_m;f]\right|\le c{\Lambda }_r(x_0,\dots ,x_m;{\omega }_k),$$

$${\Lambda }_r(x_0,\dots ,x_m;\phi )={\Lambda }_{0,r+1,r}(x_0,\dots ,x_m;\phi )={\int }_{d/2}^d{u}^{-2}\phi (u)du.$$

$$[x_0,\dots ,x_m;f]=\frac{[x_1,\dots ,x_{r+1};f]-[x_0,\dots ,x_r;f]}{x_m-x_0}=\frac{f^{(r)}({\theta }_1)-f^{(r)}({\theta }_2)}{r!d/2},$$

$$\frac{|f^{(r)}({\theta }_1)-f^{(r)}({\theta }_2)|}{d}\le \frac{{\omega }_1(d/2)}{d}\le {\int }_{d/2}^d\frac{{\omega }_1(u)}{u^2}dt={\Lambda }_r(x_0,\dots ,x_m;{\omega }_1).\text{qed}$$

$$\phi (t)\le t^{k-1}{\int }_t^d{u}^{-k}\omega (u)du,t\in (0,d/2],$$

$${\Lambda }_{p,q,r}(x_0,\dots ,x_m;\phi )\le 2^{k^2}{\Lambda }_r(x_0,\dots ,x_m;\omega ).$$

$$(p_{\nu },q_{\nu }):=\{\begin{array}{ll}(p_{\nu -1}-1,q_{\nu -1}),& \text{if}{x}_{q_{\nu -1}}-x_{p_{\nu -1}-1}\le x_{q_{\nu -1}+1}-x_{p_{\nu -1}},\\ (p_{\nu -1},q_{\nu -1}+1),& \text{otherwise}.\end{array}$$

$$q_{\nu }-p_{\nu }=q-p+\nu ,$$

$$0\le p_{\nu }\le p_{\nu -1}<q_{\nu -1}\le q_{\nu }\le m.$$

$$(p_{m-q+p},q_{m-q+p})=(0,m).$$

$$d_{\nu }:=d(p_{\nu },q_{\nu }),0\le \nu \le m-q+p.$$

$$d_{\nu }\ge d_{\nu -1}=x_{q_{\nu }}-x_{p_{\nu }}1\le \nu \le m+q-p,$$

Full text

For example, one of the numerous corollaries of this estimate is the fact that, given a function $f\in C^{(r)}(I)$ and a set $Z=\{z_{j}\}_{j=0}^{\mu}$ such that $z_{j+1}-z_{j}\geq\lambda|I|$, for all $0\leq j\leq\mu-1$, where $I:=[z_{0},z_{\mu}]$, $|I|$ is the length of $I$ and $\lambda$ is some positive number, the Hermite polynomial ${\mathcal{L}}(\cdot;f;Z)$ of degree $\leq r\mu+\mu+r$ satisfying ${\mathcal{L}}^{(j)}(z_{\nu};f;Z)=f^{(j)}(z_{\nu})$, for all $0\leq\nu\leq\mu$ and $0\leq j\leq r$, approximates $f$ so that, for all $x\in I$,

[TABLE]

where $m:=(r+1)(\mu+1)$, $C=C(m,\lambda)$ and $\mathop{\rm dist}\nolimits(x,Z):=\min_{0\leq j\leq\mu}|x-z_{j}|$.

Абстракт

Ми даємо оцiнку узагальненої роздiленої рiзницi $[x_{0},\dots,x_{m};f]$, де деякi з точок $x_{i}$ можуть спiвпадати (в цьому випадку $f$ вважається досить гладкою). Ця оцiнка потiм застосовується для суттєвого посилення вiдомих нерiвностей Уiтнi i Маршу та узагальнює їх для полiномiальної iнтерполяцiї Ермiта.

Наприклад, одним з численних наслiдкiв цiєї оцiнки є той факт, що для заданої функцiї $f\in C^{(r)}(I)$ та набору точок $Z=\{z_{j}\}_{j=0}^{\mu}$ таких, що $z_{j+1}-z_{j}\geq\lambda|I|$, для всiх $0\leq j\leq\mu-1$, де $I:=[z_{0},z_{\mu}]$, $|I|$ є довжиною $I$ та $\lambda$ є деяким додатнiм числом, полiном Ермiта ${\mathcal{L}}(\cdot;f;Z)$ степеня $\leq r\mu+\mu+r$, який задовольняє ${\mathcal{L}}^{(j)}(z_{\nu};f;Z)=f^{(j)}(z_{\nu})$, для $0\leq\nu\leq\mu$ та $0\leq j\leq r$, наближує $f$ так, що, для всiх $x\in I$,

[TABLE]

де $m:=(r+1)(\mu+1)$, $C=C(m,\lambda)$ та $\mathop{\rm dist}\nolimits(x,Z):=\min_{0\leq j\leq\mu}|x-z_{j}|$.

1 Introduction

V. K. Dzyadyk had a significant impact on the theory of extension of functions, and we start this note with recalling three of his most significant results (in our opinion) in this direction.

First, in 1956 (see [Dz56]), he solved a problem posed by S. M. Nikolskii on extending a function $f\in{\rm Lip}_{M}(\alpha,p)$, $0<\alpha\leq 1$, $p\geq 1$, on a finite interval $[a,b]$, to a function $F\in{\rm Lip}_{M_{1}}(\alpha,p)$ on the whole real line, i.e., $F|_{[a,b]}=f$.

Then, in 1958 (see [Dz58] or [Dz]*p. 171-172), he showed that if $f\in C[0,1]$ then this function may be extended to a function $F\in C[-1,1]$ with a controlled second modulus of smoothness on $[-1,1]$, i.e., $F|_{[0,1]}=f$, and the second moduli of smoothness of $f$ and $F$ satisfy $\omega_{2}(F,\delta;[-1,1])\leq 5\omega_{2}(f,\delta;[0,1])$, $0<\delta\leq 1$. (This result was independently proved by Frey [F] the same year.)

In this note, we mostly deal with results related to Dzyadyk’s third result which we will now describe.

Given a function $f\in C[a,b]$ and $a\leq x_{0}<x_{1}<x_{2}\leq b$, the second divided difference $[x_{0},x_{1},x_{2};f]$ can be estimated as follows (see, *e.g. [Dz]**p. 176 and [DS]*p. 237):

[TABLE]

where $c=\mathop{\rm const}\nolimits<18$, $h:=\min\{x_{1}-x_{0},x_{2}-x_{1}\}$.

Now, let $\omega_{2}$ be an arbitrary function of the second modulus of smoothness type, i.e., $\omega_{2}\in C[0,\infty]$ is nondecreasing and such that $\omega_{2}(0)=0$ and $t_{1}^{-2}\omega_{2}(t_{1})\leq 4t_{2}^{-2}\omega_{2}(t_{2})$, $0<t_{2}<t_{1}$.

In 1983, Dzyadyk and Shevchuk [DS83] proved that if $f$, defined on an arbitrary set $E\subset\mathbb{R}$, satisfies (1.1) with $\omega_{2}(t)$ instead of $\omega_{2}(f,t)$ for each triple of points $x_{0},x_{1},x_{2}\in E$ satisfying $x_{0}<x_{1}<x_{2}$, then $f$ may be extended from $E$ to a function $F\in C(\mathbb{R})$ such that $\omega_{2}(F,t;\mathbb{R})\leq c\omega_{2}(t)$. In other words, (1.1) with $\omega_{2}(t)$ instead of $\omega_{2}(f,t)$ is necessary and sufficient for a function $f$ to be the trace, on the set $E\subset\mathbb{R}$, of a function $F\in C(\mathbb{R})$ satisfying $\omega_{2}(F,t;\mathbb{R})\leq c\omega_{2}(t)$. This result was independently proved by Brudnyi and Shvartsman [BS] in 1982 (see also Jonsson [J] for $\omega_{2}(t)=t$).

V. K. Dzyadyk posed the question to describe such traces for functions of the $k$th modulus of smoothness type with $k>2$. He conjectured that an analog of (1.1) must be a corollary of Whitney and Marchaud inequalities. In 1984, this conjecture was confirmed by Shevchuk in [Sh84pre], and a corresponding (exact) analog of (1.1) for $k>2$ was found (see (2.7) below with $r=0$). Earlier, the case $\omega(t)=t^{k-1}$ was proved by Jonsson whose paper [J] was submitted in 1981, revised in 1983 and published in 1985.

So what happens when we have differentiable functions? In 1934, Whitney [W] described the traces of $r$ times continuously differentiable functions $F:\mathbb{R}\mapsto\mathbb{R}$ on arbitrary closed sets $E\subset\mathbb{R}$: this trace consists of all functions $f:E\mapsto\mathbb{R}$ whose $r$th differences converge on $E$ (see [W-diff] for the definition). In 1975, de Boor [dB75] described the traces of functions $F:\mathbb{R}\mapsto\mathbb{R}$ with bounded $r$-th derivative on arbitrary sets $E\subset\mathbb{R}$ of isolated points: this trace consists of all functions whose $r$-th divided differences are uniformly bounded on $E$ (in 1965, Subbotin [Su] obtained exact constants in the case when sets $E$ consist of equidistant points).

Finally, given an arbitrary set $E\subset\mathbb{R}$, the necessary and sufficient condition for a function $f$ to be a trace (on $E$) of a function $F\in C^{(r)}(\mathbb{R})$ with a prescribed $k$-th modulus of continuity of the $r$-th derivative was obtained by Shevchuk in 1984 in [Sh84pre]; see also [S]*Theorems 11.1 and 12.3, [DS]*Theorems 3.2 and 4.3 in Chapter 4 and [SZ], where a linear extension operator was given.

In fact, this necessary and sufficient condition is an analog of (1.1) for the $k$-th modulus of continuity of the $r$-th derivative of $f$ which is inequality (2.7) in Theorem 2.2 below. However, the original proof of Theorem 2.2 was distributed among several publications (see [Sh84pre, Sh84, Galan] as well as [S] and [DS]), and there was an unfortunate misprint in the formulation of [DS]*Theorem 6.4 in Section 3: in (3.6.36), ‘‘$k$’’ was written instead of ‘‘$m$’’. Hence, the main purpose of this note is to properly formulate this theorem (Theorem 2.2), provide its complete self-contained proof and discuss several important corollaries/applications that have been inadvertently overlooked in the past.

2 Definitions, notations and the main result

For $f\in C[a,b]$ and any $k\in\mathbb{N}$, set

[TABLE]

and denote by

[TABLE]

the $k$th modulus of smoothness of $f$ on $[a,b]$.

Now, we recall the definition of Lagrange-Hermite divided differences (see *e.g. [DL]**p. 118). Let $X=\{x_{j}\}_{j=0}^{m}$ be a collection of $m+1$ points with possible repetitions. For each $j$, the multiplicity $m_{j}$ of $x_{j}$ is the number of $x_{i}$ such that $x_{i}=x_{j}$, and let $l_{j}$ be the number of $x_{i}=x_{j}$ with $i\leq j$. We say that a point $x_{j}$ is a simple knot if its multiplicity is $1$. Suppose that a real valued function $f$ is defined at all points in $X$ and, moreover, for each $x_{j}\in X$, $f^{(l_{j}-1)}(x_{j})$ is defined as well (i.e., $f$ has $m_{j}-1$ derivatives at each point that has multiplicity $m_{j}$).

Denote

[TABLE]

the divided difference of $f$ of order [math] at the point $x_{0}$.

Definition 2.1**.**

Let $m\in\mathbb{N}$. If $x_{0}=\dots=x_{m}$, then we denote

[TABLE]

Otherwise, $x_{0}\neq x_{j^{*}}$, for some number $j^{*}$, and we denote

[TABLE]

the divided (Lagrange-Hermite) difference of $f$ of order $m$ at the knots $X=\{x_{j}\}_{j=0}^{m}$.

Note that $[x_{0},\dots,x_{m};f]$ is symmetric in $x_{0},\dots,x_{m}$ (*i.e., *it does not depend on how the points from $X$ are numbered), and recall that

[TABLE]

is the (Hermite) polynomial of degree $\leq m$ that satisfies

[TABLE]

Hence, in particular, if $x_{j_{*}}$ is a simple knot, then we can write

[TABLE]

From now on, for convenience, we assume that all interpolation points are numbered from left to right, *i.e., *the set of interpolation points $X=\{x_{j}\}_{j=0}^{m}$ is such that $x_{0}\leq x_{1}\leq\dots\leq x_{m}$. We also assume that the maximum multiplicity of each point is $r+1$ with $r\in\mathbb{N}_{0}$, so that

[TABLE]

Also, let

[TABLE]

and note that ${\mathcal{Q}}_{m,r}=\emptyset$ if $m\leq r$.

Now, for all $(p,q)\in{\mathcal{Q}}_{m,r}$, put

[TABLE]

where $x_{-1}:=x_{0}-(x_{m}-x_{0})$ and $x_{m+1}:=x_{m}+(x_{m}-x_{0})$. Note, in particular, that

[TABLE]

Everywhere below, $\Phi$ is the set of nondecreasing functions $\varphi\in C[0,\infty]$ satisfying $\varphi(0)=0$. We also denote

[TABLE]

and

[TABLE]

Here, we use the usual convention that $\prod_{i=0}^{-1}:=1$ and $\prod_{i=m+1}^{m}:=1$.

The following theorem is the main result of this paper.

Theorem 2.2**.**

Let $r\in\mathbb{N}_{0}$ and $m\in\mathbb{N}$ be such that $m\geq r+1$, and suppose that a set $X=\{x_{j}\}_{j=0}^{m}$ is such that $x_{0}\leq x_{1}\leq\dots\leq x_{m}$ and (2.5) is satisfied. If $f\in C^{(r)}[x_{0},x_{m}]$, then

[TABLE]

where $k:=m-r$ and $\omega_{k}(t):=\omega_{k}(f^{(r)},t;[x_{0},x_{m}])$, and the constant $c$ depends only on $m$.

3 Auxiliary lemmas

Throughout this section, we assume that $r\in\mathbb{N}_{0}$, $m\in\mathbb{N}$, $m\geq r+1$, the set $X=\{x_{j}\}_{j=0}^{m}$ is such that $x_{0}\leq x_{1}\leq\dots\leq x_{m}$ and (2.5) is satisfied, and that $(p,q)\in{\mathcal{Q}}_{m,r}$. For convenience, we also denote $k:=m-r$.

We first show that Theorem 2.2 is valid in the case $m=r+1$ (i.e., $k=1$).

Lemma 3.1**.**

Theorem 2.2 holds if $m=r+1$.

Proof.

If $m=r+1$, then ${\mathcal{Q}}_{m,r}=\{(0,r+1)\}$, and so

[TABLE]

Hence, since $x_{0}\neq x_{m}$ by assumption (2.5), (2.7) follows from the identity

[TABLE]

where $\theta_{1}\in(x_{1},x_{r+1})$ and $\theta_{2}\in(x_{0},x_{r})$, and the estimate

[TABLE]

∎

For $k>2$, we need the following lemma.

Lemma 3.2**.**

Let $(p,q)\in{\mathcal{Q}}_{m,r}$ be such that $q-p+2\leq m$. If $\varphi\in\Phi$ and $\omega\in\Phi$ are such that

[TABLE]

then

[TABLE]

Proof.

Let $(p,q)\in{\mathcal{Q}}_{m,r}$ such that $q-p+2\leq m$ be fixed, and consider the collection $\{(p_{\nu},q_{\nu})\}_{\nu=0}^{m-q+p}$ which we define as follows. Let $(p_{0},q_{0}):=(p,q)$, and for $\nu\geq 1$,

[TABLE]

It is clear that $q_{\nu}-p_{\nu}=q_{\nu-1}-p_{\nu-1}+1$, and so

[TABLE]

and one can easily check (for example, by induction) that, for all $1\leq\nu\leq m-q+p$,

[TABLE]

Hence, in particular,

[TABLE]

In the rest of this proof, we use the notation

[TABLE]

Also, observe that

[TABLE]

and

[TABLE]

We now show that, for all $1\leq\nu\leq m-q+p$,

[TABLE]

Indeed, if $x_{q_{\nu-1}}-x_{p_{\nu-1}-1}\leq x_{q_{\nu-1}+1}-x_{p_{\nu-1}}$, then $(p_{\nu},q_{\nu})=(p_{\nu-1}-1,q_{\nu-1})$, $d_{\nu-1}=x_{q_{\nu-1}}-x_{p_{\nu-1}-1}$ and, for $q_{\nu-1}+1\leq j\leq m$,

[TABLE]

whence

[TABLE]

that yields (3.4) because $m-q_{\nu-1}\leq m-q\leq k$.

Similarly, if $x_{q_{\nu-1}}-x_{p_{\nu-1}-1}>x_{q_{\nu-1}+1}-x_{p_{\nu-1}}$, then $(p_{\nu},q_{\nu})=(p_{\nu-1},q_{\nu-1}+1)$, $d_{\nu-1}=x_{q_{\nu-1}+1}-x_{p_{\nu-1}}$, and, for $0\leq j\leq p_{\nu-1}-1$,

[TABLE]

and whence

[TABLE]

that also yields (3.4) because $p_{\nu-1}\leq p<k$.

Inequality (3.4) implies that, for all $1\leq\nu\leq m-q+p$,

[TABLE]

It is clear that $d(p,q)\leq x_{m}-x_{0}=d/2$, and so condition (3.1) implies that

[TABLE]

Using integration by parts we write

[TABLE]

The last estimate is obvious for $1\leq\nu\leq m-q+p-1$ and, for $\mu=m-q+p$, it follows from

[TABLE]

which is valid because

[TABLE]

Finally, taking into account (3.3), (3.5) and recalling that $d_{\nu-1}=x_{q_{\nu}}-x_{p_{\nu}}$, $1\leq\nu\leq m-q+p$, we obtain

[TABLE]

that implies (3.2). ∎

Lemma 3.3**.**

If $k=m-r\geq 2$ and $\varphi\in\Phi$ and $\omega\in\Phi$ are such that

[TABLE]

and $\varphi(t)\leq\omega(t)$, $t\in[d/2,d]$, then

[TABLE]

and

[TABLE]

where constants $c$ depend only on $k$.

Proof.

We first note that (3.8) is a consequence of (3.7). Indeed, given $X=\{x_{i}\}_{i=0}^{m}$, define the set $Y=\{y_{i}\}_{i=0}^{m}$ by letting $y_{i}:=-x_{m-i}$, $0\leq i\leq m$. Then, $y_{0}\leq y_{1}\leq\dots\leq y_{m}$, $y_{m}-y_{0}=x_{m}-x_{0}$ (and so, in particular, $d(Y)=d(X)=d$),

[TABLE]

and it is not difficult to check that, for any $\psi\in\Phi$,

[TABLE]

and

[TABLE]

Hence, using the fact that $(p,q)\in{\mathcal{Q}}_{\mu,r}$ iff $(\mu-q,\mu-p)\in{\mathcal{Q}}_{\mu,r}$, $\mu=m-1,m$ we have

[TABLE]

and

[TABLE]

and so (3.8) follows from (3.7) applied to the set $Y$.

We are now ready to prove (3.7). Let $(p^{*},q^{*})\in{\mathcal{Q}}_{m-1,r}$ be such that

[TABLE]

and denote, for convenience, $X_{m}:=\{x_{0},\dots,x_{m}\}$ and $X_{m-1}:=\{x_{0},\dots,x_{m-1}\}$.

We consider four cases.

Case I: $(p^{*},q^{*})=(0,m-1)$.

We put $h:=x_{m-1}-x_{0}$ and note that $\displaystyle\Lambda^{*}=\int_{h}^{2h}u^{-k}\varphi(u)du$.

If $h\leq d/4$, then

[TABLE]

If $h>d/4$, then

[TABLE]

Case II: either (i) $q^{*}\neq m-1$, or (ii) $q^{*}=m-1$, $p^{*}>0$, and $x_{m}-x_{p^{*}}>x_{m-1}-x_{p^{*}-1}$

In this case, $d(p^{*},q^{*};X_{m-1})=d(p^{*},q^{*};X_{m})=x_{m-1}-x_{p^{*}-1}$, and so

[TABLE]

Since $q^{*}-p^{*}+2\leq m$, we may apply Lemma 3.2 and obtain (3.7).

Case III: $q^{*}=m-1$, $p^{*}\geq 2$ and $x_{m}-x_{p^{*}}\leq x_{m-1}-x_{p^{*}-1}$

In this case, $d(p^{*},q^{*};X_{m-1})=x_{m-1}-x_{p^{*}-1}$ and $d(p^{*},q^{*};X_{m})=x_{m}-x_{p^{*}}$. Hence, taking into account that, for $0\leq i\leq{p^{*}}-1$,

[TABLE]

we have

[TABLE]

Since $m-p^{*}+2\leq m$, we may apply Lemma 3.2 to obtain (3.7).

Case IV: $(p^{*},q^{*})=(1,m-1)$ and $x_{m}-x_{1}\leq x_{m-1}-x_{0}$

In this case, we have

[TABLE]

Now,

[TABLE]

and

[TABLE]

∎

4 Proof of Theorem 2.2

Proof.

We use induction on $k=m-r$. The base case $k=1$ is addressed in Lemma 3.1. Suppose now that $k\geq 2$ is given, assume that Theorem 2.2 holds for $k-1$ and prove it for $k$.

Denote by $P_{k-1}$ the polynomial of best uniform approximation of $f^{(r)}$ on $[x_{0},x_{m}]$ of degree at most $k-1$, and let $g$ be such that

[TABLE]

Then

[TABLE]

and Whitney’s inequality yields

[TABLE]

Hence, the well known Marchaud inequality:

if $F\in C[a,b]$ and $1\leq\ell<k$, then, for all

[TABLE]

implies, for $0<t\leq x_{m}-x_{0}$,

[TABLE]

We also note that (4.1) implies, in particular, that for all $t\in[x_{m}-x_{0},2(x_{m}-x_{0})]$,

[TABLE]

We now represent the divided difference in the form

[TABLE]

where $y_{j}:=x_{j+1}$, $0\leq j\leq m-1$. By the induction hypothesis,

[TABLE]

and

[TABLE]

Now, taking into account (4.2), (4.3) and homogeneity of $\Lambda_{r}(z_{0},\dots,z_{m};\psi)$ with respect to $\psi$, Lemma 3.3 with $\varphi:=\omega_{k-1}^{g}$ and $\omega:=K\omega_{k}^{f}$, where $K$ is the maximum of constants $c$ in (4.2) and (4.3), implies that

[TABLE]

and

[TABLE]

which yields (2.7). ∎

5 Applications

Throughout this section, the set $X=\{x_{j}\}_{j=0}^{m-1}$ is assumed to be such that $x_{0}\leq x_{1}\leq\dots\leq x_{m-1}$ (unless stated otherwise), and denote $I:=[x_{0},x_{m-1}]$ and $|I|=x_{m-1}-x_{0}$. Also, all constants written in the form $C(\mu_{1},\mu_{2},\dots)$ may depend only on parameters $\mu_{1}$, $\mu_{2}$, … and not on anything else.

We first recall that the classical Whitney interpolation inequality can be written in the following form.

Theorem 5.1** (Whitney inequality, [W57]).**

Let $r\in\mathbb{N}_{0}$ and $m\in\mathbb{N}$ be such that $m\geq\max\{r+1,2\}$, and suppose that a set $X=\{x_{j}\}_{j=0}^{m-1}$ is such that

[TABLE]

where $0<\lambda\leq 1$. If $f\in C^{(r)}(I)$, then

[TABLE]

where $L_{m-1}(\cdot;f;x_{0},\dots,x_{m-1})$ is the (Lagrange) polynomial of degree $\leq m-1$ interpolating $f$ at the points in $X$.

We emphasize that condition (5.1) implies that the points in the set $X$ in the above theorem are assumed to be sufficiently separated from one another. A natural question is what happens if condition (5.1) is not satisfied and, moreover, if some of the points in $X$ are allowed to coalesce. In that case, $L_{m-1}(\cdot;f;x_{0},\dots,x_{m-1})$ is the Hermite polynomial whose derivatives interpolate corresponding derivatives of $f$ at points that have multiplicities more than $1$, and Theorem 5.1 provides no information on its error of approximation of $f$.

It turns out that one can use Theorem 2.2 to provide an answer to this question and significantly strengthen Theorem 5.1. As far as we know the formulation of the following theorem (which is itself a corollary of a more general Theorem 5.3 below) is new and has not appeared anywhere in the literature.

Theorem 5.2**.**

Let $r\in\mathbb{N}_{0}$ and $m\in\mathbb{N}$ be such that $m\geq r+2$, and suppose that a set $X=\{x_{j}\}_{j=0}^{m-1}$ is such that

[TABLE]

where $0<\lambda\leq 1$. If $f\in C^{(r)}(I)$, then

[TABLE]

where $L_{m-1}(\cdot;f;x_{0},\dots,x_{m-1})$ is the Hermite polynomial defined in (2.2) and (2.3).

Theorem 5.2 is an immediate corollary of the following more general theorem. Before we state it, we need to introduce the following notation. Given $X=\{x_{j}\}_{j=0}^{m-1}$ with $x_{0}\leq x_{1}\leq\dots\leq x_{m-1}$ and $x\in[x_{0},x_{m-1}]$, we renumber all points $x_{j}$’s so that their distance from $x$ is nondecreasing. In other words, let $\sigma=(\sigma_{0},\dots,\sigma_{m-1})$ be a permutation of $(0,\dots,m-1)$ such that

[TABLE]

Note that this permutation $\sigma$ depends on $x$ and is not unique if there are at least two points from $X$ which are equidistant from $x$. Denote also

[TABLE]

Theorem 5.3**.**

Let $r\in\mathbb{N}_{0}$ and $m\in\mathbb{N}$ be such that $m\geq r+2$, and suppose that a set $X=\{x_{j}\}_{j=0}^{m-1}$ is such that

[TABLE]

where $0<\lambda\leq 1$. If $f\in C^{(r)}(I)$, then, for each $x\in I$,

[TABLE]

where ${\mathcal{D}}_{r}(x,X)$ is defined in (5.4), and $L_{m-1}(\cdot;f;x_{0},\dots,x_{m-1})$ is the Hermite polynomial defined in (2.2) and (2.3).

Before proving Theorem 5.3 we state another corollary. First, if $k\in\mathbb{N}$ and $\mathrm{w}(t):=\omega_{k}(f^{(r)},t;I)$, then $t_{2}^{-k}\mathrm{w}(t_{2})\leq 2^{k}t_{1}^{-k}\mathrm{w}(t_{1})$, for $0<t_{1}<t_{2}$. Hence, denoting $\lambda_{x}:=|I|\sqrt[k]{|x-x_{\sigma_{r}}|/|I|}$ and noting that $|x-x_{\sigma_{r}}|\leq\lambda_{x}\leq|I|$, we have, for $k\geq 2$,

[TABLE]

Therefore, we immediately get the following consequence of Theorem 5.3.

Corollary 5.4**.**

Let $r\in\mathbb{N}_{0}$ and $m\in\mathbb{N}$ be such that $m\geq r+2$, and suppose that a set $X=\{x_{j}\}_{j=0}^{m-1}$ is such that condition (5.5) is satisfied.

If $f\in C^{(r)}(I)$, then, for each $x\in I$,

[TABLE]

where \lambda_{x}:=|I|\big{(}|x-x_{\sigma_{r}}|/|I|\big{)}^{1/(m-r)}.

We are now ready to prove Theorem 5.3.

Proof of Theorem 5.3.

We note that all constants $C$ below may depend only on $m$ and $\lambda$ and are different even if they appear in the same line. It is clear that we can assume that $x$ is different from all $x_{j}$’s. So we let $1\leq i\leq m-1$ and $x\in(x_{i-1},x_{i})$ be fixed, and denote

[TABLE]

$Y:=\{y_{j}\}_{j=0}^{m}$, $d(Y):=2(y_{m}-y_{0})=2(x_{m-1}-x_{0})=2|I|$, $k:=m-r$, and ${\omega}_{k}(t):={\omega}_{k}(f^{(r)},t,[y_{0},y_{m}])={\omega}_{k}(f^{(r)},t,I)$. Condition (5.5) implies that $y_{j}<y_{j+r+1}$, for all $0\leq j\leq m-r-1$, and so we can use Theorem 2.2 to estimate \big{|}[y_{0},\dots,y_{m};f]\big{|}. Now, identity (2.4) with $j_{*}:=i$ that yields $y_{j_{*}}=x$ implies

[TABLE]

We also note that it is possible to show that $\prod_{j=0}^{m-1}|x-x_{j}|\geq(\lambda/2)^{k-1}{\mathcal{D}}_{r}(x,X)|I|^{k-1}$, and so the above estimate cannot be improved.

In order to estimate $\Lambda_{r}$, we suppose that $(p,q)\in{\mathcal{Q}}_{m,r}$ and estimate $\Lambda_{p,q,r}$. Since $q-p\geq r+1$, we have

[TABLE]

and

[TABLE]

Hence,

[TABLE]

We consider the following two cases.

Case 1: $q\geq p+r+2$, or $q=p+r+1$ and $x\not\in[y_{p},y_{q}]$

It is clear that $y_{q}-y_{p}\geq\lambda|I|$, and so it follows from (5.9) that

[TABLE]

Case 2: $q=p+r+1$ and $x\in[y_{p},y_{q}]$

If $x=y_{p}$, then $p=i$, $q=i+r+1$, and $y_{q}-y_{p}=x_{i+r}-x\geq|x-x_{\sigma_{r}}|$.

If $x=y_{q}$, then $q=i$, $p=i-r-1$, and $y_{q}-y_{p}=x-x_{i-r-1}\geq|x-x_{\sigma_{r}}|$.

If $x\in(y_{p},y_{q})$, then $y_{q}-y_{p}=x_{p+r}-x_{p}$. Since it is impossible that $|x-x_{\sigma_{r}}|>\max\{x-x_{p},x_{p+r}-x\}$, for this would imply that $\{p,\dots,p+r\}\subset\{\sigma_{0},\dots,\sigma_{r-1}\}$ which cannot happen since these sets have cardinalities $r+1$ and $r$, respectively, we conclude that $|x-x_{\sigma_{r}}|\leq\max\{x-x_{p},x_{p+r}-x\}\leq x_{p+r}-x_{p}$. Thus, in this case, (5.9) implies that

[TABLE]

Hence,

[TABLE]

which together with (5.8) implies (5.6). ∎

We state one more corollary to illustrate the power of Theorem 5.3. Suppose that $Z=\left\{z_{j}\right\}_{j=0}^{\mu}$ with $z_{0}<z_{1}<\dots<z_{\mu}$, and let $X=\{x_{j}\}_{j=0}^{m-1}$ with $m:=(r+1)(\mu+1)$ be such that $x_{\nu(r+1)+j}:=z_{\nu}$, for all $0\leq\nu\leq\mu$ and $0\leq j\leq r$. In other words,

[TABLE]

Now, given $f\in C^{(r)}[z_{0},z_{\mu}]$, let ${\mathcal{L}}(x;f;Z):=L_{m-1}(x,f;x_{0},\dots,x_{m-1})$ be the Hermite polynomial of degree $\leq m-1=r\mu+\mu+r$ satisfying

[TABLE]

Also,

[TABLE]

Corollary 5.5**.**

Let $r\in\mathbb{N}_{0}$ and $\mu\in\mathbb{N}$, and suppose that a set $Z=\left\{z_{j}\right\}_{j=0}^{\mu}$ is such that

[TABLE]

where $0<\lambda\leq 1$, $I:=[z_{0},z_{\mu}]$ and $|I|:=z_{\mu}-z_{0}$. If $f\in C^{(r)}(I)$, then, for each $x\in I$,

[TABLE]

where $m:=(r+1)(\mu+1)$, $C=C(m,\lambda)$ and the polynomial ${\mathcal{L}}(\cdot;f;Z)$ of degree $\leq m-1$ satisfies (5.10).

As a final note, we remark that some of the results that appeared in the literature follow from the results in this note. For example, (i) the main theorem in [Gopmz] immediately follows from Corollary 5.5 with $\mu=1$, $z_{0}=-1$ and $z_{1}=1$, (ii) Corollary 5.4 is much stronger than the main theorem in [Gop], (iii) a particular case of [K-sim]*Lemmas 8 and 9 for $k=0$ follows from Corollary 5.4, (iv) several propositions in the unconstrained case in [GLSW] follow from Corollary 5.4, (v) [LP]*Lemma 3.3, Corollaries 3.4-3.6 follow from Corollary 5.4, and (vi) the proof of [kls]*Lemma 3.1 may be simplified if Corollary 5.4 is used.

{bibsection} de BoorC.How small can one make the derivatives of an interpolating function?Collection of articles dedicated to G. G. Lorentz on the occasion of his sixty-fifth birthday, IIJ. Approximation Theory131975105–116; erratum, ibid. 14 (1975), 82@article{dB75, author = {de Boor, C.}, title = {How small can one make the derivatives of an interpolating function?}, note = {Collection of articles dedicated to G. G. Lorentz on the occasion of his sixty-fifth birthday, II}, journal = {J. Approximation Theory}, volume = {13}, date = {1975}, pages = {105–116; erratum, ibid. 14 (1975), 82}} BrudnyiYu. A.ShvartsmanP. A.Description of the trace of a function from the generalized lipschitz space to an arbitrary compact setRussiantitle={Studies in the theory of functions of several real variables (Russian)}, series={Matematika}, publisher={Yaroslav. Gos. Univ., Yaroslavl'}, 198216–24@article{BS, author = {Brudnyi, Yu. A.}, author = {Shvartsman, P. A.}, title = {Description of the trace of a function from the generalized Lipschitz space to an arbitrary compact set}, language = {Russian}, conference = {title={Studies in the theory of functions of several real variables (Russian)}, }, book = {series={Matematika}, publisher={Yaroslav. Gos. Univ., Yaroslavl'}, }, date = {1982}, pages = {16–24}} DeVoreR. A.LorentzG. G.Constructive approximationA Series of Comprehensive Studies in Mathematics303Springer-VerlagNew York1993x+227@book{DL, author = {DeVore, R. A.}, author = {Lorentz, G. G.}, title = {Constructive approximation}, series = {A Series of Comprehensive Studies in Mathematics}, volume = {303}, publisher = {Springer-Verlag}, place = {New York}, date = {1993}, pages = {x+227}} DzyadykV. K.Continuation of functions satisfying a lipschitz condition in the $L_{p}$ metricRussianMat. Sb. N.S.40(82)1956239–242@article{Dz56, author = {Dzyadyk, V. K.}, title = {Continuation of functions satisfying a Lipschitz condition in the $L_p$ metric}, language = {Russian}, journal = {Mat. Sb. N.S.}, volume = {40(82)}, date = {1956}, pages = {239–242}} DzyadykV. K.A further strengthening of jackson’s theorem on the approximation of continuous functions by ordinary polynomialsRussianDokl. Akad. Nauk SSSR1211958403–406ISSN 0002-3264@article{Dz58, author = {Dzyadyk, V. K.}, title = {A further strengthening of Jackson's theorem on the approximation of continuous functions by ordinary polynomials}, language = {Russian}, journal = {Dokl. Akad. Nauk SSSR}, volume = {121}, date = {1958}, pages = {403–406}, issn = {0002-3264}} DzyadykV. K.Introduction to the theory of uniform approximation of functions by polynomialsRussianIzdat. ‘‘Nauka’’, Moscow1977511@book{Dz, author = {Dzyadyk, V. K.}, title = { Introduction to the Theory of Uniform Approximation of Functions by Polynomials}, language = {Russian}, publisher = {Izdat. ``Nauka'', Moscow}, date = {1977}, pages = {511}} DzyadykV. K.ShevchukI. A.Continuation of functions which, on an arbitrary set of the line, are traces of functions with a given second modulus of continuityRussianIzv. Akad. Nauk SSSR Ser. Mat.4719832248–267@article{DS83, author = {Dzyadyk, V. K.}, author = {Shevchuk, I. A.}, title = {Continuation of functions which, on an arbitrary set of the line, are traces of functions with a given second modulus of continuity}, language = {Russian}, journal = {Izv. Akad. Nauk SSSR Ser. Mat.}, volume = {47}, date = {1983}, number = {2}, pages = {248–267}} DzyadykV. K.ShevchukI. A.Theory of uniform approximation of functions by polynomialsWalter de GruyterBerlin2008xv+480@book{DS, author = {Dzyadyk, V. K.}, author = {Shevchuk, I. A.}, title = {Theory of Uniform Approximation of Functions by Polynomials}, publisher = {Walter de Gruyter}, place = {Berlin}, date = {2008}, pages = {xv+480}} FreyT.On local best approximation by polynomials. iiHungarianMagyar Tud. Akad. Mat. Fiz. Oszt. Közl.8195889–112@article{F, author = {Frey, T.}, title = {On local best approximation by polynomials. II}, language = {Hungarian}, journal = {Magyar Tud. Akad. Mat. Fiz. Oszt. K"{o}zl.}, volume = {8}, date = {1958}, pages = {89–112}} GalanV. D.Smooth functions and estimates for derivativesRussianCandidate of Science Thesis – Kyiv, Ukraine1991@book{Galan, author = {Galan, V. D.}, title = {Smooth functions and estimates for derivatives}, language = {Russian}, note = {Candidate of Science Thesis – Kyiv, Ukraine}, date = {1991}} GonskaH. H.LeviatanD.ShevchukI. A.WenzH.-J.Interpolatory pointwise estimates for polynomial approximationConstr. Approx.1620004603–629@article{GLSW, author = {Gonska, H. H.}, author = {Leviatan, D.}, author = {Shevchuk, I. A.}, author = {Wenz, H.-J.}, title = {Interpolatory pointwise estimates for polynomial approximation}, journal = {Constr. Approx.}, volume = {16}, date = {2000}, number = {4}, pages = {603–629}} GopengauzI. E.A pointwise error estimate for interpolation with multiple nodes at endpoints of an intervalRussianMat. Zametki511992155–61journal={Math. Notes}, volume={51}, date={1992}, number={1-2}, pages={36–40}, @article{Gopmz, author = {Gopengauz, I. E.}, title = {A pointwise error estimate for interpolation with multiple nodes at endpoints of an interval}, language = {Russian}, journal = {Mat. Zametki}, volume = {51}, date = {1992}, number = {1}, pages = {55–61}, translation = {journal={Math. Notes}, volume={51}, date={1992}, number={1-2}, pages={36–40}, }} GopengauzI. E.Pointwise estimates of the hermitian interpolationJ. Approx. Theory771994131–41@article{Gop, author = {Gopengauz, I. E.}, title = {Pointwise estimates of the Hermitian interpolation}, journal = {J. Approx. Theory}, volume = {77}, date = {1994}, number = {1}, pages = {31–41}} JonssonA.The trace of the zygmund class $\Lambda_{k}(R)$ to closed sets and interpolating polynomialsJ. Approx. Theory44198511–13@article{J, author = {Jonsson, A.}, title = {The trace of the Zygmund class $\Lambda_k(R)$ to closed sets and interpolating polynomials}, journal = {J. Approx. Theory}, volume = {44}, date = {1985}, number = {1}, pages = {1–13}} KopotunK. A.Simultaneous approximation by algebraic polynomialsConstr. Approx.121996167–94@article{K-sim, author = {Kopotun, K. A.}, title = {Simultaneous approximation by algebraic polynomials}, journal = {Constr. Approx.}, volume = {12}, date = {1996}, number = {1}, pages = {67–94}} KopotunK. A.LeviatanD.ShevchukI. A.Interpolatory estimates for convex piecewise polynomial approximation(submitted)https://arxiv.org/pdf/1811.01087.pdf@article{kls, author = {Kopotun, K. A.}, author = {Leviatan, D.}, author = {Shevchuk, I. A.}, title = {Interpolatory estimates for convex piecewise polynomial approximation}, journal = {(submitted)}, eprint = {https://arxiv.org/pdf/1811.01087.pdf}} LeviatanD.PetrovaI. L.Interpolatory estimates in monotone piecewise polynomial approximationJ. Approx. Theory22320171–8Corrigendum, J. Approx. Theory 228 (2018), 79–80@article{LP, author = {Leviatan, D.}, author = {Petrova, I. L.}, title = {Interpolatory estimates in monotone piecewise polynomial approximation}, journal = {J. Approx. Theory}, volume = {223}, date = {2017}, pages = {1–8}, language = {Corrigendum, J. Approx. Theory {\bf 228} (2018), 79–80}} ShevchukI. A.Extension of functions, which are traces of functions belonging to $H_{k}^{\varphi}$ on an arbitrary subset of the lineEnglish, with Russian summaryAnal. Math.1019843249–273@article{Sh84, author = {Shevchuk, I. A.}, title = {Extension of functions, which are traces of functions belonging to $H_k^\varphi$ on an arbitrary subset of the line}, language = {English, with Russian summary}, journal = {Anal. Math.}, volume = {10}, date = {1984}, number = {3}, pages = {249–273}} ShevchukI. A.Constructive description of traces of differentiable functions of a real variableRussianAkad. Nauk Ukrain. SSR Inst. Mat. Preprint19841940@article{Sh84pre, author = {Shevchuk, I. A.}, title = {Constructive description of traces of differentiable functions of a real variable}, language = {Russian}, journal = {Akad. Nauk Ukrain. SSR Inst. Mat. Preprint}, date = {1984}, number = {19}, pages = {40}} ShevchukI. A.Polynomial approximation and traces of functions continuous on a segmentNaukova Dumka, KievRussian1992@book{S, author = {Shevchuk, I. A.}, title = {Polynomial approximation and traces of functions continuous on a segment}, publisher = {Naukova Dumka, Kiev}, language = {Russian}, date = {1992}} ShevchukI. A.ZhelnovO. D.Linear bounded operator for extension of traces of differentiable functions on $\mathbb{R}$East J. Approx.1020041-2133–158@article{SZ, author = {Shevchuk, I. A.}, author = {Zhelnov, O. D.}, title = {Linear bounded operator for extension of traces of differentiable functions on $\R$}, journal = {East J. Approx.}, volume = {10}, date = {2004}, number = {1-2}, pages = {133–158}} SubbotinJu. N.On the connection between finite differences and corresponding derivativesRussianTrudy Mat. Inst. Steklov.78196524–42@article{Su, author = {Subbotin, Ju. N.}, title = {On the connection between finite differences and corresponding derivatives}, language = {Russian}, journal = {Trudy Mat. Inst. Steklov.}, volume = {78}, date = {1965}, pages = {24–42}} WhitneyH.Analytic extensions of differentiable functions defined in closed setsTrans. Amer. Math. Soc.361934163–89@article{W, author = {Whitney, H.}, title = {Analytic extensions of differentiable functions defined in closed sets}, journal = {Trans. Amer. Math. Soc.}, volume = {36}, date = {1934}, number = {1}, pages = {63–89}} WhitneyH.Differentiable functions defined in closed sets. iTrans. Amer. Math. Soc.3619342369–387@article{W-diff, author = {Whitney, H.}, title = {Differentiable functions defined in closed sets. I}, journal = {Trans. Amer. Math. Soc.}, volume = {36}, date = {1934}, number = {2}, pages = {369–387}} WhitneyH.On functions with bounded $n^{th}$ differencesJ. Math. Pure et Appl.36195767–95@article{W57, author = {Whitney, H.}, title = {On functions with bounded $n^{th}$ differences}, journal = {J. Math. Pure et Appl.}, volume = {36}, date = {1957}, pages = {67–95}}