On one estimate of divided differences and its applications
We give an estimate of the general divided differences $[x_0, ..., x_m; f]$, where some points xi are allowed to coalesce (in this case, $f$ is assumed to be sufficiently smooth). This estimate is then applied to significantly strengthen the celebrated Whitney and Marchaud inequalities and their gen...
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2019
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507184119414784 |
|---|---|
| author | Kopotun, K. A. Leviatan, D. Shevchuk, I. A. Копотун, К. А. Левіатан, Д. Шевчук, І. О. |
| author_facet | Kopotun, K. A. Leviatan, D. Shevchuk, I. A. Копотун, К. А. Левіатан, Д. Шевчук, І. О. |
| author_sort | Kopotun, K. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T08:54:43Z |
| description | We give an estimate of the general divided differences $[x_0, ..., x_m; f]$, where some points xi are allowed to coalesce (in this case, $f$ is assumed to be sufficiently smooth). This estimate is then applied to significantly strengthen the celebrated
Whitney and Marchaud inequalities and their generalization to the 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 $Z = \{ z_j\}^{\mu}_{j=0}$ 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 a positive number, the Hermite polynomial $\scrL (\cdot ; f;Z)$ of degree $\leq r\mu + \mu + r$ satisfying the equality
$\scrL (j)(z\nu ; f;Z) = f(j)(z\nu )$ for all $0 \leq \nu \leq \mu$ and $0 \leq j \leq r$ approximates $f$ so that, for all $x \in I$,
$$| f(x) \scr L (x; f;Z)| \leq C (\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t} (x,Z))^{r+1}
\int^{2| I|}_{dist (x,Z)}\frac{\omega_{m-r}(f^{(r)}, t, I)}{t^2}dt,$$
where $m := (r + 1)(\mu + 1), C = C(m, \lambda )$ and $\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t} (x,Z) := \mathrm{m}\mathrm{i}\mathrm{n}0\leq j\leq \mu | x zj | $. |
| first_indexed | 2026-03-24T02:05:16Z |
| format | Article |
| fulltext |
UDC 517.5
K. A. Kopotun (Univ. Manitoba, Canada),
D. Leviatan (Raymond and Beverly Sackler School Math. Sci., Tel Aviv Univ., Israel),
I. A. Shevchuk (Taras Shevchenko Nat. Univ. Kyiv, Ukraine)
ON ONE ESTIMATE OF DIVIDED DIFFERENCES AND ITS APPLICATIONS*
ПРО ОДНУ ОЦIНКУ ДЛЯ ПОДIЛЕНИХ РIЗНИЦЬ ТА ЇЇ ЗАСТОСУВАННЯ
We give an estimate of the general divided differences [x0, . . . , xm; f ], where some points xi are allowed to coalesce (in
this case, f is assumed to be sufficiently smooth). This estimate is then applied to significantly strengthen the celebrated
Whitney and Marchaud inequalities and their generalization to the 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 = \{ zj\} \mu j=0 such that zj+1 - zj \geq \lambda | I| for all 0 \leq j \leq \mu - 1, where I := [z0, z\mu ], | I| is the length of
I, and \lambda is a positive number, the Hermite polynomial \scrL (\cdot ; f ;Z) of degree \leq r\mu + \mu + r satisfying the equality
\scrL (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,
| f(x) - \scrL (x; f ;Z)| \leq C (\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t} (x, Z))r+1
2| I| \int
dist (x,Z)
\omega m - r(f
(r), t, I)
t2
dt,
where m := (r + 1)(\mu + 1), C = C(m,\lambda ) and \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t} (x, Z) := \mathrm{m}\mathrm{i}\mathrm{n}0\leq j\leq \mu | x - zj | .
Наведено оцiнку узагальненої подiленої рiзницi [x0, . . . , xm; f ], де деякi з точок xi можуть збiгатися (в цьому
випадку f вважається досить гладкою). Цю оцiнку потiм застосовано для суттєвого посилення вiдомих нерiвностей
Уiтнi i Маршу та узагальнення їх для полiномiальної iнтерполяцiї Ермiта.
Наприклад, одним iз численних наслiдкiв цiєї оцiнки є той факт, що для заданої функцiї f \in C(r)(I) та набору
точок Z = \{ zj\} \mu j=0 таких, що zj+1 - zj \geq \lambda | I| для всiх 0 \leq j \leq \mu - 1, де I := [z0, z\mu ], | I| — довжина I, \lambda —
деяке додатне число, полiном Ермiта \scrL (\cdot ; f ;Z) степеня \leq r\mu + \mu + r , який задовольняє \scrL (j)(z\nu ; f ;Z) = f (j)(z\nu )
для 0 \leq \nu \leq \mu i 0 \leq j \leq r, наближає f так, що для всiх x \in I
| f(x) - \scrL (x; f ;Z)| \leq C (\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t} (x, Z))r+1
2| I| \int
dist (x,Z)
\omega m - r(f
(r), t, I)
t2
dt,
де m := (r + 1)(\mu + 1), C = C(m,\lambda ) i \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t} (x, Z) := \mathrm{m}\mathrm{i}\mathrm{n}0\leq j\leq \mu | x - zj | .
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 [4]), he solved a problem posed by S. M. Nikolskii on extending a function
f \in \mathrm{L}\mathrm{i}\mathrm{p}M (\alpha , p), 0 < \alpha \leq 1, p \geq 1, on a finite interval [a, b], to a function F \in \mathrm{L}\mathrm{i}\mathrm{p}M1
(\alpha , p) on the
whole real line, i.e., F | [a,b] = f.
Then, in 1958 (see [5] or [6, 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
\leq 5\omega 2(f, \delta ; [0, 1]), 0 < \delta \leq 1. (This result was independently proved by Frey [9] the same year.)
In this note, we mostly deal with results related to Dzyadyk’s third result which we will now
describe.
* Supported by NSERC of Canada.
c\bigcirc K. A. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK, 2019
230 ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
ON ONE ESTIMATE OF DIVIDED DIFFERENCES AND ITS APPLICATIONS 231
Given a function f \in C[a, b] and a \leq x0 < x1 < x2 \leq b, the second divided difference
[x0, x1, x2; f ] can be estimated as follows (see, e.g., [16, p. 176] and [8, p. 237]):
| [x0, x1, x2; f ]| \leq
c
x2 - x0
x2 - x0\int
h
\omega 2(f, t)
t2
dt, (1.1)
where c = \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t} < 18, h := \mathrm{m}\mathrm{i}\mathrm{n}\{ x1 - x0, x2 - x1\} .
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 - 2
1 \omega 2(t1) \leq 4t - 2
2 \omega 2(t2), 0 < t2 < t1.
In 1983, Dzyadyk and Shevchuk [7] proved that if f, defined on an arbitrary set E \subset \BbbR , satisfies
(1.1) with \omega 2(t) instead of \omega 2(f, t) for each triple of points x0, x1, x2 \in E satisfying x0 < x1 < x2,
then f may be extended from E to a function F \in C(\BbbR ) such that \omega 2(F, t;\BbbR ) \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 \BbbR , of a function F \in C(\BbbR ) satisfying \omega 2(F, t;\BbbR ) \leq c\omega 2(t). This result was
independently proved by Brudnyi and Shvartsman [2] in 1982 (see also Jonsson [14] for \omega 2(t) = t).
V. K. Dzyadyk posed the question to describe such traces for functions of the kth 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 [19], 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) = tk - 1 was proved by Jonsson whose paper [14] was submitted in 1981, revised in 1983
and published in 1985.
So what happens when we have differentiable functions? In 1934, Whitney [23] described the
traces of r times continuously differentiable functions F : \BbbR \mapsto \rightarrow \BbbR on arbitrary closed sets E \subset \BbbR :
this trace consists of all functions f : E \mapsto \rightarrow \BbbR whose rth differences converge on E (see [24] for
the definition). In 1975, de Boor [1] described the traces of functions F : \BbbR \mapsto \rightarrow \BbbR with bounded rth
derivative on arbitrary sets E \subset \BbbR of isolated points: this trace consists of all functions whose rth
divided differences are uniformly bounded on E (in 1965, Subbotin [22] obtained exact constants in
the case when sets E consist of equidistant points).
Finally, given an arbitrary set E \subset \BbbR , the necessary and sufficient condition for a function f to
be a trace (on E ) of a function F \in C(r)(\BbbR ) with a prescribed kth modulus of continuity of the
rth derivative was obtained by Shevchuk in 1984 in [19] (see also Theorems 11.1 and 12.3 in [20],
Theorems 3.2 and 4.3 in Chapter 4 of [8], and [21], where a linear extension operator was given).
In fact, this necessary and sufficient condition is an analog of (1.1) for the kth modulus of
continuity of the rth derivative of f which is inequality (2.7) in Theorem 2.1 below. However, the
original proof of Theorem 2.1 was distributed among several publications (see [10, 18, 19] as well as
[20] and [8]), and there was an unfortunate misprint in the formulation of Theorem 6.4 in Section 3
of [8]: in (3.6.36), “k” was written instead of “m”. Hence, the main purpose of this note is to
properly formulate this theorem (Theorem 2.1), provide its complete self-contained proof and discuss
several important corollaries/applications that have been inadvertently overlooked in the past.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
232 K. A. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
2. Definitions, notations and the main result. For f \in C[a, b] and any k \in \BbbN , set
\Delta k
u(f, x; [a, b]) :=
\left\{
\sum k
i=0
( - 1)i
\biggl(
k
i
\biggr)
f(x+ (k/2 - i)u), x\pm (k/2)u \in [a, b],
0, otherwise,
and denote by
\omega k(f, t; [a, b]) := \mathrm{s}\mathrm{u}\mathrm{p}
0<u\leq t
\| \Delta k
u(f, \cdot ; [a, b])\| C[a,b] (2.1)
the kth modulus of smoothness of f on [a, b].
Now, we recall the definition of Lagrange – Hermite divided differences (see, e.g., [3, p. 118]). Let
X = \{ xj\} mj=0 be a collection of m+ 1 points with possible repetitions. For each j, the multiplicity
mj of xj is the number of xi such that xi = xj , and let lj be the number of xi = xj with i \leq j.
We say that a point xj 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 xj \in X, f (lj - 1)(xj) is defined as well (i.e., f
has mj - 1 derivatives at each point that has multiplicity mj ).
Denote
[x0; f ] := f(x0),
the divided difference of f of order 0 at the point x0.
Definition 2.1. Let m \in \BbbN . If x0 = . . . = xm, then we denote
[x0, . . . , xm; f ] = [x0, . . . , x0\underbrace{} \underbrace{}
m+1
; f ] :=
f (m)(x0)
m!
.
Otherwise, x0 \not = xj\ast , for some number j\ast , and we denote
[x0, . . . , xm; f ] :=
1
xj\ast - x0
([x1, . . . , xm; f ] - [x0, . . . , xj\ast - 1, xj\ast +1, . . . , xm; f ]) ,
the divided (Lagrange – Hermite) difference of f of order m at the knots X = \{ xj\} mj=0.
Note that [x0, . . . , xm; f ] is symmetric in x0, . . . , xm (i.e., it does not depend on how the points
from X are numbered), and recall that
Lm(x; f) := Lm(x; f ;x0, . . . , xm) := f(x0) +
m\sum
j=1
[x0, . . . , xj ; f ](x - x0) . . . (x - xj - 1) (2.2)
is the (Hermite) polynomial of degree \leq m that satisfies
L
(lj - 1)
m (xj ; f) = f (lj - 1)(xj), for all 0 \leq j \leq m. (2.3)
Hence, in particular, if xj\ast is a simple knot, then we can write
[x0, . . . , xm; f ] :=
f(xj\ast ) - Lm - 1(xj\ast ; f ;x0, . . . , xj\ast - 1, xj\ast +1, . . . , xm)\prod m
j=0,j \not =j\ast
(xj\ast - xj)
. (2.4)
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 = \{ xj\} mj=0 is such that x0 \leq x1 \leq . . . \leq xm. We also
assume that the maximum multiplicity of each point is r + 1 with r \in \BbbN 0, so that
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
ON ONE ESTIMATE OF DIVIDED DIFFERENCES AND ITS APPLICATIONS 233
xj < xj+r+1, for all 0 \leq j \leq m - r - 1. (2.5)
Also, let
\scrQ m,r := \{ (p, q) | 0 \leq p, q \leq m and q - p \geq r + 1\} =
= \{ (p, q) | 0 \leq p \leq m - r - 1 and p+ r + 1 \leq q \leq m\} , (2.6)
and note that \scrQ m,r = \varnothing if m \leq r.
Now, for all (p, q) \in \scrQ m,r, put
d(p, q) := d(p, q;X) := \mathrm{m}\mathrm{i}\mathrm{n}\{ xq+1 - xp, xq - xp - 1\} ,
where x - 1 := x0 - (xm - x0) and xm+1 := xm + (xm - x0). Note, in particular, that
d := d(X) := d(0,m;X) = 2(xm - x0).
Everywhere below, \Phi is the set of nondecreasing functions \varphi \in C[0,\infty ] satisfying \varphi (0) = 0.
We also denote
\Lambda p,q,r(x0, . . . , xm;\varphi ) :=
\int d(p,q)
xq - xp
up+r - q - 1\varphi (u)du\prod p - 1
i=0
(xq - xi)
\prod m
i=q+1
(xi - xp)
, (p, q) \in \scrQ m,r,
and
\Lambda r(x0, . . . , xm;\varphi ) := \mathrm{m}\mathrm{a}\mathrm{x}
(p,q)\in \scrQ m,r
\Lambda p,q,r(x0, . . . , xm;\varphi ).
Here, we use the usual convention that
\prod - 1
i=0
:= 1 and
\prod m
i=m+1
:= 1.
The following theorem is the main result of this paper.
Theorem 2.1. Let r \in \BbbN 0 and m \in \BbbN be such that m \geq r + 1, and suppose that a set
X = \{ xj\} mj=0 is such that x0 \leq x1 \leq . . . \leq xm and (2.5) is satisfied. If f \in C(r)[x0, xm], then
| [x0, . . . , xm; f ]| \leq c\Lambda r(x0, . . . , xm;\omega k), (2.7)
where k := m - r and \omega k(t) := \omega k(f
(r), t; [x0, xm]), and the constant c depends only on m.
3. Auxiliary lemmas. Throughout this section, we assume that r \in \BbbN 0, m \in \BbbN , m \geq r + 1,
the set X = \{ xj\} mj=0 is such that x0 \leq x1 \leq . . . \leq xm and (2.5) is satisfied, and that (p, q) \in \scrQ m,r.
For convenience, we also denote k := m - r.
We first show that Theorem 2.1 is valid in the case m = r + 1 (i.e., k = 1).
Lemma 3.1. Theorem 2.1 holds if m = r + 1.
Proof. If m = r + 1, then \scrQ m,r = \{ (0, r + 1)\} , and so
\Lambda r(x0, . . . , xm;\varphi ) = \Lambda 0,r+1,r(x0, . . . , xm;\varphi ) =
d\int
d/2
u - 2\varphi (u)du.
Hence, since x0 \not = xm by assumption (2.5), (2.7) follows from the identity
[x0, . . . , xm; f ] =
[x1, . . . , xr+1; f ] - [x0, . . . , xr; f ]
xm - x0
=
f (r)(\theta 1) - f (r)(\theta 2)
r!d/2
,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
234 K. A. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
where \theta 1 \in (x1, xr+1) and \theta 2 \in (x0, xr), and the estimate\bigm| \bigm| f (r)(\theta 1) - f (r)(\theta 2)
\bigm| \bigm|
d
\leq \omega 1(d/2)
d
\leq
d\int
d/2
\omega 1(u)
u2
dt = \Lambda r(x0, . . . , xm;\omega 1).
Lemma 3.1 is proved.
For k > 2, we need the following lemma.
Lemma 3.2. Let (p, q) \in \scrQ m,r be such that q - p+ 2 \leq m. If \varphi \in \Phi and \omega \in \Phi are such that
\varphi (t) \leq tk - 1
d\int
t
u - k\omega (u)du, t \in (0, d/2], (3.1)
then
\Lambda p,q,r(x0, . . . , xm;\varphi ) \leq 2k
2
\Lambda r(x0, . . . , xm;\omega ). (3.2)
Proof. Let (p, q) \in \scrQ m,r such that q - p + 2 \leq m be fixed, and consider the collection
\{ (p\nu , q\nu )\} m - q+p
\nu =0 which we define as follows. Let (p0, q0) := (p, q) and, for \nu \geq 1,
(p\nu , q\nu ) :=
\left\{ (p\nu - 1 - 1, q\nu - 1), if xq\nu - 1 - xp\nu - 1 - 1 \leq xq\nu - 1+1 - xp\nu - 1 ,
(p\nu - 1, q\nu - 1 + 1), otherwise.
It is clear that q\nu - p\nu = q\nu - 1 - p\nu - 1 + 1, and so
q\nu - p\nu = q - p+ \nu , (3.3)
and one can easily check (for example, by induction) that, for all 1 \leq \nu \leq m - q + p,
0 \leq p\nu \leq p\nu - 1 < q\nu - 1 \leq q\nu \leq m.
Hence, in particular,
(pm - q+p, qm - q+p) = (0,m).
In the rest of this proof, we use the notation
d\nu := d(p\nu , q\nu ), 0 \leq \nu \leq m - q + p.
Also, observe that
d\nu \geq d\nu - 1 = xq\nu - xp\nu , 1 \leq \nu \leq m+ q - p,
and
dm - q+p - 1 = xm - x0 = d/2.
We now show that, for all 1 \leq \nu \leq m - q + p,
d\nu - 1\prod p\nu - 1 - 1
i=0
(xq\nu - 1 - xi)
\prod m
i=q\nu - 1+1
(xi - xp\nu - 1)
\leq
\leq 2k\prod p\nu - 1
i=0
(xq\nu - xi)
\prod m
i=q\nu +1
(xi - xp\nu )
. (3.4)
Indeed, if xq\nu - 1 - xp\nu - 1 - 1 \leq xq\nu - 1+1 - xp\nu - 1 , then (p\nu , q\nu ) = (p\nu - 1 - 1, q\nu - 1), d\nu - 1 = xq\nu - 1 -
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
ON ONE ESTIMATE OF DIVIDED DIFFERENCES AND ITS APPLICATIONS 235
- xp\nu - 1 - 1 and, for q\nu - 1 + 1 \leq j \leq m,
xj - xp\nu = (xj - xq\nu - 1) + (xq\nu - 1 - xp\nu - 1 - 1) \leq
\leq (xj - xp\nu - 1) + (xq\nu - 1+1 - xp\nu - 1) \leq 2(xj - xp\nu - 1),
whence
m\prod
i=q\nu - 1+1
(xi - xp\nu - 1) \geq 2q\nu - 1 - m
m\prod
i=q\nu +1
(xi - xp\nu ),
that yields (3.4) because m - q\nu - 1 \leq m - q \leq k.
Similarly, if xq\nu - 1 - xp\nu - 1 - 1 > xq\nu - 1+1 - xp\nu - 1 , then (p\nu , q\nu ) = (p\nu - 1, q\nu - 1 + 1), d\nu - 1 =
= xq\nu - 1+1 - xp\nu - 1 , and, for 0 \leq j \leq p\nu - 1 - 1,
xq\nu - xj = (xq\nu - 1+1 - xp\nu - 1) + (xp\nu - 1 - xj) <
< (xq\nu - 1 - xp\nu - 1 - 1) + (xq\nu - 1 - xj) \leq 2(xq\nu - 1 - xj),
and whence
p\nu - 1 - 1\prod
i=0
(xq\nu - 1 - xi) \geq 2 - p\nu - 1
p\nu - 1\prod
i=0
(xq\nu - xi),
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,\prod \nu - 1
i=0
di\prod p - 1
i=0
(xq - xi)
\prod m
i=q+1
(xi - xp)
\leq 2k\nu \prod p\nu - 1
i=0
(xq\nu - xi)
\prod m
i=q\nu +1
(xi - xp\nu )
. (3.5)
It is clear that d(p, q) \leq xm - x0 = d/2, and so condition (3.1) implies that
d(p,q)\int
xq - xp
up+r - q - 1\varphi (u)du \leq
d(p,q)\int
xq - xp
up+m - q - 2
\left( d\int
u
v - k\omega (v)dv
\right) du.
Using integration by parts we write
(m - q + p - 1)
d(p,q)\int
xq - xp
up+r - q - 1\varphi (u)du -
d(p,q)\int
xq - xp
up+r - q - 1\omega (u)du \leq
\leq dm - q+p - 1(p, q)
d\int
d(p,q)
\omega (u)
uk
du = dm - q+p - 1(p, q)
m - q+p\sum
\nu =1
d\nu \int
d\nu - 1
\omega (u)
uk
du \leq
\leq 2
m - q+p\sum
\nu =1
\nu - 1\prod
i=0
di
d\nu \int
d\nu - 1
up+r - q - 1 - \nu \omega (u)du.
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236 K. A. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
The last estimate is obvious for 1 \leq \nu \leq m - q + p - 1 and, for \mu = m - q + p, it follows from
dm - q+p - 1
0 dm - q - p \leq 2
m - q+p - 1\prod
i=0
di
which is valid because
dm - q+p - 1
0 \leq
m - q+p - 2\prod
i=0
di and dm - q - p = d(0,m) = d = 2dm - q+p - 1.
Finally, taking into account (3.3), (3.5) and recalling that d\nu - 1 = xq\nu - xp\nu , 1 \leq \nu \leq m - q+ p, we
obtain
(m - q + p - 1)\Lambda p,q,r(x0, . . . , xm;\varphi ) \leq
\leq \Lambda p,q,r(x0, . . . , xm;\omega ) + 2
m - q+p\sum
\nu =1
2k\nu \Lambda p\nu ,q\nu ,r(x0, . . . , xm;\omega )
that implies (3.2).
Lemma 3.2 is proved.
Lemma 3.3. If k = m - r \geq 2 and \varphi \in \Phi and \omega \in \Phi are such that
\varphi (t) \leq tk - 1
d\int
t
u - k\omega (u)du, t \in (0, d/2], (3.6)
and \varphi (t) \leq \omega (t), t \in [d/2, d], then
\Lambda r(x0, . . . , xm - 1;\varphi ) \leq c(xm - x0)\Lambda r(x0, . . . , xm;\omega ) (3.7)
and
\Lambda r(x1, . . . , xm;\varphi ) \leq c(xm - x0)\Lambda r(x0, . . . , xm;\omega ), (3.8)
where constants c depend only on k.
Proof. We first note that (3.8) is a consequence of (3.7). Indeed, given X = \{ xi\} mi=0, define the
set Y = \{ yi\} mi=0 by letting yi := - xm - i, 0 \leq i \leq m. Then y0 \leq y1 \leq . . . \leq ym, ym - y0 = xm - x0
(and so, in particular, d(Y ) = d(X) = d),
d(p, q;Y ) = \mathrm{m}\mathrm{i}\mathrm{n}\{ yq+1 - yp, yq - yp - 1\} = \mathrm{m}\mathrm{i}\mathrm{n}\{ xm - p - xm - q - 1, xm - p+1 - xm - q\} =
= d(m - q,m - p;X) = d(m - q,m - p),
and it is not difficult to check that, for any \psi \in \Phi ,
\Lambda p,q,r(y0, . . . , ym;\psi ) = \Lambda m - q,m - p,r(x0, . . . , xm;\psi )
and
\Lambda p,q,r(y0, . . . , ym - 1;\psi ) = \Lambda m - q - 1,m - p - 1,r(x1, . . . , xm;\psi ).
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ON ONE ESTIMATE OF DIVIDED DIFFERENCES AND ITS APPLICATIONS 237
Hence, using the fact that (p, q) \in \scrQ \mu ,r iff (\mu - q, \mu - p) \in \scrQ \mu ,r, \mu = m - 1,m, we have
\Lambda r(x0, . . . , xm;\omega ) = \mathrm{m}\mathrm{a}\mathrm{x}
(p,q)\in \scrQ m,r
\Lambda p,q,r(x0, . . . , xm;\omega ) =
= \mathrm{m}\mathrm{a}\mathrm{x}
(m - q,m - p)\in \scrQ m,r
\Lambda m - q,m - p,r(y0, . . . , ym;\omega ) = \Lambda r(y0, . . . , ym;\omega )
and
\Lambda r(x1, . . . , xm;\varphi ) = \mathrm{m}\mathrm{a}\mathrm{x}
(p,q)\in \scrQ m - 1,r
\Lambda p,q,r(x1, . . . , xm;\varphi ) =
= \mathrm{m}\mathrm{a}\mathrm{x}
(m - q - 1,m - p - 1)\in \scrQ m - 1,r
\Lambda m - q - 1,m - p - 1,r(y0, . . . , ym - 1;\varphi ) = \Lambda r(y0, . . . , ym - 1;\varphi ),
and so (3.8) follows from (3.7) applied to the set Y.
We are now ready to prove (3.7). Let (p\ast , q\ast ) \in \scrQ m - 1,r be such that
\Lambda \ast := \Lambda p\ast ,q\ast ,r(x0, . . . , xm - 1;\varphi ) = \Lambda r(x0, . . . , xm - 1;\varphi ),
and denote, for convenience, Xm := \{ x0, . . . , xm\} and Xm - 1 := \{ x0, . . . , xm - 1\} .
We consider four cases.
Case I: (p\ast , q\ast ) = (0,m - 1).
We put h := xm - 1 - x0 and note that \Lambda \ast =
\int 2h
h
u - k\varphi (u)du.
If h \leq d/4, then
21 - k\Lambda \ast \leq (2h)1 - k\varphi (2h) \leq
d\int
2h
u - k\omega (u)du \leq
d/2\int
h
u - k\omega (u)du+
d\int
d/2
u - k\omega (u)du \leq
\leq
d/2\int
h
u - k\omega (u)du+ d
d\int
d/2
u - k - 1\omega (u)du =
= (xm - x0) (\Lambda 0,m - 1,r(x0, . . . , xm;\omega ) + 2\Lambda 0,m,r(x0, . . . , xm;\omega )) \leq
\leq 3(xm - x0)\Lambda r(x0, . . . , xm;\omega ).
If h > d/4, then
\Lambda \ast =
d/2\int
h
u - k\varphi (u)du+
2h\int
d/2
u - k\varphi (u)du \leq (4/d)k - 1 \varphi (d/2) +
2h\int
d/2
u - k\varphi (u)du <
< 4k
d\int
d/2
u - k\varphi (u)du \leq 4k
d\int
d/2
u - k\omega (u)du \leq 4kd
d\int
d/2
u - k - 1\omega (u)du =
= 2 \cdot 4k(xm - x0)\Lambda 0,m,r(x0, . . . , xm;\omega ) \leq 2 \cdot 4k(xm - x0)\Lambda r(x0, . . . , xm;\omega ).
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238 K. A. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
Case II: either (i) q\ast \not = m - 1, or (ii) q\ast = m - 1, p\ast > 0, and xm - xp\ast > xm - 1 - xp\ast - 1.
In this case, d (p\ast , q\ast ;Xm - 1) = d (p\ast , q\ast ;Xm) = xm - 1 - xp\ast - 1, and so
\Lambda \ast = (xm - xp\ast ) \Lambda p\ast ,q\ast ,r(x0, . . . , xm;\varphi ) \leq (xm - x0)\Lambda p\ast ,q\ast ,r(x0, . . . , xm;\varphi ).
Since q\ast - p\ast + 2 \leq m, we may apply Lemma 3.2 and obtain (3.7).
Case III: q\ast = m - 1, p\ast \geq 2 and xm - xp\ast \leq xm - 1 - xp\ast - 1.
In this case, d (p\ast , q\ast ;Xm - 1) = xm - 1 - xp\ast - 1 and d (p\ast , q\ast ;Xm) = xm - xp\ast . Hence, taking
into account that, for 0 \leq i \leq p\ast - 1,
xm - xi = xm - xp\ast + xp\ast - xi \leq xm - 1 - xp\ast - 1 + xp\ast - xi \leq 2(xm - 1 - xi),
we get
\Lambda p\ast ,m - 1,r(x0, . . . , xm - 1;\varphi ) - (xm - xp\ast ) \Lambda p\ast ,m - 1,r(x0, . . . , xm;\varphi ) =
=
p\ast - 1\prod
i=0
(xm - 1 - xi)
- 1
xm - 1 - xp\ast - 1\int
xm - xp\ast
up
\ast +r - m\varphi (u)du \leq
\leq 2p
\ast
p\ast - 1\prod
i=0
(xm - xi)
- 1(xm - xp\ast - 1)
xm - xp\ast - 1\int
xm - xp\ast
up
\ast +r - m - 1\varphi (u)du =
= 2p
\ast
(xm - xp\ast - 1) \Lambda p\ast ,m,r(x0, . . . , xm;\varphi ).
Since m - p\ast + 2 \leq m, we may apply Lemma 3.2 to obtain (3.7).
Case IV: (p\ast , q\ast ) = (1,m - 1) and xm - x1 \leq xm - 1 - x0.
In this case, we have
\Lambda \ast =
1
xm - 1 - x0
xm - 1 - x0\int
xm - 1 - x1
u1 - k\varphi (u)du \leq
\leq 1
xm - 1 - x0
xm - 1 - x0\int
xm - 1 - x1
\left( d\int
u
v - k\omega (v)dv
\right) du \leq
\leq
d\int
xm - 1 - x0
u - k\omega (u) du+
1
xm - 1 - x0
xm - 1 - x0\int
xm - 1 - x1
u1 - k\omega (u)du =: \scrA 1 +\scrA 2.
Now,
\scrA 1 =
d/2\int
xm - 1 - x0
u - k\omega (u)du+
d\int
d/2
u - k\omega (u)du \leq
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ON ONE ESTIMATE OF DIVIDED DIFFERENCES AND ITS APPLICATIONS 239
\leq
d/2\int
xm - 1 - x0
u - k\omega (u)du+ d
d\int
d/2
u - k - 1\omega (u)du =
= (xm - x0) (\Lambda 0,m - 1,r(x0, . . . , xm;\omega ) + 2\Lambda 0,m,r(x0, . . . , xm;\omega )) \leq
\leq 3(xm - x0)\Lambda r(x0, . . . , xm;\omega )
and
\scrA 2 =
1
xm - 1 - x0
xm - x1\int
xm - 1 - x1
u1 - k\omega (u)du+
1
xm - 1 - x0
xm - 1 - x0\int
xm - x1
u1 - k\omega (u)du \leq
\leq (xm - x1)\Lambda 1,m - 1,r(x0, . . . , xm;\omega ) +
xm - 1 - x0\int
xm - x1
u - k\omega (u)du \leq
\leq (xm - x0)\Lambda 1,m - 1,r(x0, . . . , xm;\omega ) +
xm - x0\int
xm - x1
u - k\omega (u)du =
= (xm - x0) (\Lambda 1,m - 1,r(x0, . . . , xm;\omega ) + \Lambda 1,m,r(x0, . . . , xm;\omega )) \leq
\leq 2(xm - x0)\Lambda r(x0, . . . , xm;\omega ).
Lemma 3.3 is proved.
4. Proof of Theorem 2.1. 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 2.1 holds for k - 1 and prove
it for k.
Denote by Pk - 1 the polynomial of best uniform approximation of f (r) on [x0, xm] of degree at
most k - 1, and let g be such that
g(r) := f (r) - Pk - 1.
Then
\omega k(g
(r), t; [x0, xm]) = \omega k(f
(r), t; [x0, xm]) =: \omega f
k (t),
and Whitney’s inequality yields\bigm\| \bigm\| \bigm\| g(r)\bigm\| \bigm\| \bigm\|
[x0,xm]
\leq c\omega k
\Bigl(
f (r), xm - x0; [x0, xm]
\Bigr)
= c\omega f
k (xm - x0). (4.1)
Hence, the well known Marchaud inequality:
if F \in C[a, b] and 1 \leq \ell < k, then, for all 0 < t \leq b - a,
\omega \ell (F, t; [a, b]) \leq c(k)t\ell
\left( b - a\int
t
\omega k(F, u; [a, b])
u\ell +1
du+
\| F\| [a,b]
(b - a)\ell
\right) ,
implies, for 0 < t \leq xm - x0,
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240 K. A. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
\omega g
k - 1(t) := \omega k - 1
\Bigl(
g(r), t; [x0, xm]
\Bigr)
\leq
\leq ctk - 1
\left( xm - x0\int
t
\omega f
k (u)
uk
du+
\omega f
k (xm - x0)
(xm - x0)k - 1
\right) \leq
\leq ctk - 1
2(xm - x0)\int
t
\omega f
k (u)
uk
du. (4.2)
We also note that (4.1) implies, in particular, that, for all t \in [xm - x0, 2(xm - x0)],
\omega g
k - 1(t) \leq c\| g(r)\| [x0,xm] \leq c\omega f
k (xm - x0) \leq c\omega f
k (t). (4.3)
We now represent the divided difference in the form
(xm - x0)[x0, . . . , xm; f ] = (xm - x0)[x0, . . . , xm; g] =
= [x1, . . . , xm; g] - [x0, . . . , xm - 1; g] = [y0, . . . , ym - 1; g] - [x0, . . . , xm - 1; g],
where yj := xj+1, 0 \leq j \leq m - 1. By the induction hypothesis,
| [x0, . . . , xm - 1; g]| \leq c\Lambda r(x0, . . . , xm - 1;\omega
g
k - 1)
and
| [y0, . . . , ym - 1; g]| \leq c\Lambda r
\bigl(
y0, . . . , ym - 1;\omega
g
k - 1
\bigr)
.
Now, taking into account (4.2), (4.3) and homogeneity of \Lambda r(z0, . . . , zm;\psi ) with respect to \psi ,
Lemma 3.3 with \varphi := \omega g
k - 1 and \omega := K\omega f
k , where K is the maximum of constants c in (4.2) and
(4.3), implies that
\Lambda r
\bigl(
x0, . . . , xm - 1;\omega
g
k - 1
\bigr)
\leq c(xm - x0)\Lambda r
\Bigl(
x0, . . . , xm;\omega f
k
\Bigr)
and
\Lambda r
\bigl(
y0, . . . , ym - 1;\omega
g
k - 1
\bigr)
= \Lambda r
\bigl(
x1, . . . , xm;\omega g
k - 1
\bigr)
\leq c(xm - x0)\Lambda r
\Bigl(
x0, . . . , xm;\omega f
k
\Bigr)
,
which yields (2.7).
Theorem 2.1 is proved.
5. Applications. Throughout this section, the set X = \{ xj\} m - 1
j=0 is assumed to be such that
x0 \leq x1 \leq . . . \leq xm - 1 (unless stated otherwise), and denote I := [x0, xm - 1] and | I| = xm - 1 - x0.
Also, all constants written in the form C(\mu 1, \mu 2, . . .) 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, [25]). Let r \in \BbbN 0 and m \in \BbbN be such that m \geq \mathrm{m}\mathrm{a}\mathrm{x}\{ r +
+ 1, 2\} , and suppose that a set X = \{ xj\} m - 1
j=0 is such that
xj+1 - xj \geq \lambda | I| , for all 0 \leq j \leq m - 2, (5.1)
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ON ONE ESTIMATE OF DIVIDED DIFFERENCES AND ITS APPLICATIONS 241
where 0 < \lambda \leq 1. If f \in C(r)(I), then
| f(x) - Lm - 1(x; f ;x0, . . . , xm - 1)| \leq C(m,\lambda )| I| r\omega m - r(f
(r), | I| , I), x \in I,
where Lm - 1(\cdot ; f ;x0, . . . , xm - 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, Lm - 1(\cdot ; f ;x0, . . . , xm - 1) is the Hermite polynomial whose derivatives interpolate corre-
sponding 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 2.1 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 \BbbN 0 and m \in \BbbN be such that m \geq r + 2, and suppose that a set
X = \{ xj\} m - 1
j=0 is such that
xj+r+1 - xj \geq \lambda | I| , for all 0 \leq j \leq m - r - 2, (5.2)
where 0 < \lambda \leq 1. If f \in C(r)(I), then\bigm| \bigm| f(x) - Lm - 1(x; f ;x0, . . . , xm - 1)
\bigm| \bigm| \leq C(m,\lambda )| I| r\omega m - r(f
(r), | I| , I), x \in I,
where Lm - 1(\cdot ; f ;x0, . . . , xm - 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 = \{ xj\} m - 1
j=0 with x0 \leq x1 \leq . . . \leq xm - 1
and x \in [x0, xm - 1], we renumber all points xj ’s so that their distance from x is nondecreasing. In
other words, let \sigma = (\sigma 0, . . . , \sigma m - 1) be a permutation of (0, . . . ,m - 1) such that
| x - x\sigma \nu - 1 | \leq | x - x\sigma \nu | , for all 1 \leq \nu \leq m - 1. (5.3)
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
\scrD r(x,X) :=
r\prod
\nu =0
| x - x\sigma \nu | , 0 \leq r \leq m - 1. (5.4)
Theorem 5.3. Let r \in \BbbN 0 and m \in \BbbN be such that m \geq r + 2, and suppose that a set
X = \{ xj\} m - 1
j=0 is such that
xj+r+1 - xj \geq \lambda | I| , for all 0 \leq j \leq m - r - 2, (5.5)
where 0 < \lambda \leq 1. If f \in C(r)(I), then, for each x \in I,\bigm| \bigm| f(x) - Lm - 1(x; f ;x0, . . . , xm - 1)
\bigm| \bigm| \leq
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242 K. A. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
\leq C(m,\lambda )\scrD r(x,X)
2| I| \int
| x - x\sigma r |
\omega m - r(f
(r), t, I)
t2
dt, (5.6)
where \scrD r(x,X) is defined in (5.4), and Lm - 1(\cdot ; f ;x0, . . . , xm - 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 \BbbN and \mathrm{w}(t) := \omega k(f
(r), t; I),
then t - k
2 \mathrm{w}(t2) \leq 2kt - k
1 \mathrm{w}(t1), for 0 < t1 < t2. Hence, denoting \lambda x := | I| k
\sqrt{}
| x - x\sigma r | /| I| and noting
that | x - x\sigma r | \leq \lambda x \leq | I| , we have, for k \geq 2,
2| I| \int
| x - x\sigma r |
\mathrm{w}(t)
t2
dt =
\left( \lambda x\int
| x - x\sigma r |
+
2| I| \int
\lambda x
\right) \mathrm{w}(t)
t2
dt \leq
\leq \mathrm{w}(\lambda x)
\infty \int
| x - x\sigma r |
t - 2 dt+ 2k\lambda - k
x \mathrm{w}(\lambda x)
2| I| \int
0
tk - 2 dt =
=
\mathrm{w}(\lambda x)
| x - x\sigma r |
\biggl(
1 +
22k - 1
k - 1
\biggr)
.
Therefore, we immediately get the following consequence of Theorem 5.3.
Corollary 5.1. Let r \in \BbbN 0 and m \in \BbbN be such that m \geq r + 2, and suppose that a set
X = \{ xj\} m - 1
j=0 is such that condition (5.5) is satisfied.
If f \in C(r)(I), then, for each x \in I,\bigm| \bigm| f(x) - Lm - 1(x; f ;x0, . . . , xm - 1)
\bigm| \bigm| \leq C(m,\lambda )\scrD r - 1(x,X)\omega m - r(f
(r), \lambda x, I) \leq
\leq C(m,\lambda )\scrD r - 1(x,X)\omega m - r(f
(r), | I| , I), (5.7)
where \lambda x := | I|
\bigl(
| x - x\sigma r | /| I|
\bigr) 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 xj ’s. So we let 1 \leq i \leq m - 1 and x \in (xi - 1, xi) be fixed, and denote
yj :=
\left\{
xj , if 0 \leq j \leq i - 1,
x, if j = i,
xj - 1, if i+ 1 \leq j \leq m,
Y := \{ yj\} mj=0, d(Y ) := 2(ym - y0) = 2(xm - 1 - x0) = 2| I| , k := m - r, and \omega k(t) :=
:= \omega k
\bigl(
f (r), t, [y0, ym]
\bigr)
= \omega k
\bigl(
f (r), t, I
\bigr)
.
Condition (5.5) implies that yj < yj+r+1, for all 0 \leq j \leq m - r - 1, and so we can use
Theorem 2.1 to estimate | [y0, . . . , ym; f ]| . Now, identity (2.4) with j\ast := i that yields yj\ast = x
implies
| f(x) - Lm - 1(x; f ;x0, . . . , xm - 1)| =
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ON ONE ESTIMATE OF DIVIDED DIFFERENCES AND ITS APPLICATIONS 243
= | f(x) - Lm - 1(x; f ; y0, . . . , yi - 1, yi+1, . . . , ym)| =
= | [y0, . . . , ym; f ]|
m\prod
j=0,j \not =i
| x - yj | \leq
\leq c\Lambda r(y0, . . . , ym;\omega k)
m - 1\prod
j=0
| x - xj | \leq c\scrD r(x,X)| I| k - 1\Lambda r(y0, . . . , ym;\omega k). (5.8)
We also note that it is possible to show that
\prod m - 1
j=0
| x - xj | \geq (\lambda /2)k - 1\scrD 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 \scrQ m,r and estimate \Lambda p,q,r. Since q - p \geq r+1,
we have
yq - yi \geq yq - yp - 1 \geq yp+r+1 - yp - 1 \geq \lambda | I| , for 0 \leq i \leq p - 1,
and
yi - yp \geq yq+1 - yp \geq yp+r+2 - yp \geq \lambda | I| , for q + 1 \leq i \leq m.
Hence,
\Lambda p,q,r(y0, . . . , ym;\omega k) \leq C| I| q - m - p
2| I| \int
yq - yp
up+r - q - 1\omega k(u)du. (5.9)
We consider the two cases.
Case I: q \geq p+ r + 2, or q = p+ r + 1 and x \not \in [yp, yq].
It is clear that yq - yp \geq \lambda | I| , and so it follows from (5.9) that
\Lambda p,q,r(y0, . . . , ym;\omega k) \leq C| I| - k\omega k(| I| ) \leq C| I| 1 - k
2| I| \int
| I|
\omega k(u)
u2
du.
Case II: q = p+ r + 1 and x \in [yp, yq].
If x = yp, then p = i, q = i+ r + 1, and yq - yp = xi+r - x \geq | x - x\sigma r | .
If x = yq, then q = i, p = i - r - 1, and yq - yp = x - xi - r - 1 \geq | x - x\sigma r | .
If x \in (yp, yq), then yq - yp = xp+r - xp. Since it is impossible that | x - x\sigma r | > \mathrm{m}\mathrm{a}\mathrm{x}\{ x -
- xp, xp+r - x\} , for this would imply that \{ p, . . . , p + r\} \subset \{ \sigma 0, . . . , \sigma r - 1\} which cannot happen
since these sets have cardinalities r + 1 and r, respectively, we conclude that | x - x\sigma r | \leq \mathrm{m}\mathrm{a}\mathrm{x}\{ x -
- xp, xp+r - x\} \leq xp+r - xp. Thus, in this case, (5.9) implies that
\Lambda p,q,r(y0, . . . , ym;\omega k) \leq C| I| 1 - k
2| I| \int
| x - x\sigma r |
\omega k(u)
u2
du.
Hence,
\Lambda r(y0, . . . , ym;\omega k) \leq C| I| 1 - k
2| I| \int
| x - x\sigma r |
\omega k(u)
u2
du,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
244 K. A. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
which together with (5.8) implies (5.6).
Theorem 5.3 is proved.
We state one more corollary to illustrate the power of Theorem 5.3. Suppose that Z = \{ zj\} \mu j=0
with z0 < z1 < . . . < z\mu , and let X = \{ xj\} m - 1
j=0 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,
X =
\left\{ z0, . . . , z0\underbrace{} \underbrace{}
r+1
, z1, . . . , z1\underbrace{} \underbrace{}
r+1
, . . . , z\mu , . . . , z\mu \underbrace{} \underbrace{}
r+1
\right\} .
Now, given f \in C(r)[z0, z\mu ], let \scrL (x; f ;Z) := Lm - 1(x, f ;x0, . . . , xm - 1) be the Hermite polyno-
mial of degree \leq m - 1 = r\mu + \mu + r satisfying
\scrL (j)(z\nu ; f ;Z) = f (j)(z\nu ), for all 0 \leq \nu \leq \mu and 0 \leq j \leq r. (5.10)
Also,
\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x, Z) := \mathrm{m}\mathrm{i}\mathrm{n}
0\leq j\leq \mu
| x - zj | , x \in \BbbR .
Corollary 5.2. Let r \in \BbbN 0 and \mu \in \BbbN , and suppose that a set Z = \{ zj\} \mu j=0 is such that
zj+1 - zj \geq \lambda | I| , for all 0 \leq j \leq \mu - 1,
where 0 < \lambda \leq 1, I := [z0, z\mu ] and | I| := z\mu - z0. If f \in C(r)(I), then, for each x \in I,
\bigm| \bigm| f(x) - \scrL (x; f ;Z)
\bigm| \bigm| \leq C (\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x, Z))r+1
2| I| \int
dist(x,Z)
\omega m - r(f
(r), t, I)
t2
dt \leq
\leq C (\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x, Z))r \omega m - r
\Bigl(
f (r), | I| (\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x, Z)/| I| )1/(m - r) , I
\Bigr)
\leq
\leq C (\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x, Z))r \omega m - r(f
(r), | I| , I),
where m := (r + 1)(\mu + 1), C = C(m,\lambda ) and the polynomial \scrL (\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 [12] immediately follows from Corollary 5.2
with \mu = 1, z0 = - 1 and z1 = 1, (ii) Corollary 5.1 is much stronger than the main theorem in
[13], (iii) a particular case in Lemmas 8 and 9 of [15] for k = 0 follows from Corollary 5.1,
(iv) several propositions in the unconstrained case in [11] follow from Corollary 5.1, (v) Lemma 3.3
and Corollaries 3.4 – 3.6 of [17] follow from Corollary 5.1 and (vi) the proof of Lemma 3.1 of [16]
may be simplified if Corollary 5.1 is used.
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Received 09.11.18
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
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| id | umjimathkievua-article-1434 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:05:16Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/36/9242e1573635aefd7215f073b531e136.pdf |
| spelling | umjimathkievua-article-14342019-12-05T08:54:43Z On one estimate of divided differences and its applications Про одну оцiнку для подiлених рiзниць та її застосування Kopotun, K. A. Leviatan, D. Shevchuk, I. A. Копотун, К. А. Левіатан, Д. Шевчук, І. О. We give an estimate of the general divided differences $[x_0, ..., x_m; f]$, where some points xi are allowed to coalesce (in this case, $f$ is assumed to be sufficiently smooth). This estimate is then applied to significantly strengthen the celebrated Whitney and Marchaud inequalities and their generalization to the 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 $Z = \{ z_j\}^{\mu}_{j=0}$ 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 a positive number, the Hermite polynomial $\scrL (\cdot ; f;Z)$ of degree $\leq r\mu + \mu + r$ satisfying the equality $\scrL (j)(z\nu ; f;Z) = f(j)(z\nu )$ for all $0 \leq \nu \leq \mu$ and $0 \leq j \leq r$ approximates $f$ so that, for all $x \in I$, $$| f(x) \scr L (x; f;Z)| \leq C (\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t} (x,Z))^{r+1} \int^{2| I|}_{dist (x,Z)}\frac{\omega_{m-r}(f^{(r)}, t, I)}{t^2}dt,$$ where $m := (r + 1)(\mu + 1), C = C(m, \lambda )$ and $\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t} (x,Z) := \mathrm{m}\mathrm{i}\mathrm{n}0\leq j\leq \mu | x zj | $. Наведено оцiнку узагальненої подiленої рiзницi $[x_0, ..., x_m; f]$, де деякi з точок $x_i$ можуть збiгатися (в цьому випадку $f$ вважається досить гладкою). Цю оцiнку потiм застосовано для суттєвого посилення вiдомих нерiвностей Уiтнi i Маршу та узагальнення їх для полiномiальної iнтерполяцiї Ермiта. Наприклад, одним iз численних наслiдкiв цiєї оцiнки є той факт, що для заданої функцiї $f \in C(r)(I)$ та набору точок $Z = \{ z_j\}^{\mu}_{j=0}$ таких, що $z_{j+1} - z_j \geq \lambda | I|$ для всiх $0 \leq j \leq \mu 1$, де $I := [z_0, z_{\mu} ], | I|$ — довжина $I, \lambda $ — деяке додатне число, полiном Ермiта $\scr L(\cdot ; f;Z)$ степеня $\leq r\mu + \mu + r$, який задовольняє $\scr L^{(j)}(z\nu ; f;Z) = f(j)(z\nu )$ для $0 \leq \nu \leq \mu$ i $0 \leq j \leq r$, наближає $f$ так, що для всiх $x \in I$ $$| f(x) \scr L (x; f;Z)| \leq C (\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t} (x,Z))^{r+1} \int^{2| I|}_{dist (x,Z)}\frac{\omega_{m-r}(f^{(r)}, t, I)}{t^2}dt,$$ де $m := (r + 1)(\mu + 1), C = C(m, \lambda)$ i $\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t} (x,Z) := \mathrm{m}\mathrm{i}\mathrm{n}0\leq j\leq \mu | x zj | $. Institute of Mathematics, NAS of Ukraine 2019-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1434 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 2 (2019); 230-245 Український математичний журнал; Том 71 № 2 (2019); 230-245 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1434/418 Copyright (c) 2019 Kopotun K. A.; Leviatan D.; Shevchuk I. A. |
| spellingShingle | Kopotun, K. A. Leviatan, D. Shevchuk, I. A. Копотун, К. А. Левіатан, Д. Шевчук, І. О. On one estimate of divided differences and its applications |
| title | On one estimate of divided differences and its
applications |
| title_alt | Про одну оцiнку для подiлених рiзниць та її застосування |
| title_full | On one estimate of divided differences and its
applications |
| title_fullStr | On one estimate of divided differences and its
applications |
| title_full_unstemmed | On one estimate of divided differences and its
applications |
| title_short | On one estimate of divided differences and its
applications |
| title_sort | on one estimate of divided differences and its
applications |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1434 |
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