Are the degrees of the best (co)convex and unconstrained polynomial approximations the same? II
In Part I of the paper, we have proved that, for every $α > 0$ and a continuous function $f$, which is either convex $(s = 0)$ or changes convexity at a finite collection $Y_s = \{y_i\}^s_i = 1$ of points $y_i ∈ (-1, 1)$, $$\sup \left\{n^{\alpha}E^{(2)}_n(f,Y_s):\;n \geq N^{*}\right\} \leq c(...
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| Date: | 2010 |
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Institute of Mathematics, NAS of Ukraine
2010
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| author | Kopotun, K. A. Leviatan, D. Shevchuk, I. A. Коротун, К. А. Левіатан, Д. Шевчук, І. О. |
| author_facet | Kopotun, K. A. Leviatan, D. Shevchuk, I. A. Коротун, К. А. Левіатан, Д. Шевчук, І. О. |
| author_sort | Kopotun, K. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2020-03-18T19:39:19Z |
| description | In Part I of the paper, we have proved that, for every $α > 0$ and a continuous function $f$, which is either convex $(s = 0)$ or changes convexity at a finite collection $Y_s = \{y_i\}^s_i = 1$ of points $y_i ∈ (-1, 1)$,
$$\sup \left\{n^{\alpha}E^{(2)}_n(f,Y_s):\;n \geq N^{*}\right\} \leq c(\alpha,s) \sup \left\{n^{\alpha}E_n(f):\; n \geq 1 \right\},$$
where $E_n (f)$ and $E^{(2)}_n (f, Y_s)$ denote, respectively, the degrees of the best unconstrained and (co)convex approximations and $c(α, s)$ is a constant depending only on $α$ and $s$. Moreover, it has been shown that $N^{∗}$ may be chosen to be 1 for $s = 0$ or $s = 1, α ≠ 4$, and that it must depend on $Y_s$ and $α$ for $s = 1, α = 4$ or $s ≥ 2$.
In Part II of the paper, we show that a more general inequality
$$\sup \left\{n^{\alpha}E^{(2)}_n(f,Y_s):\;n \geq N^{*}\right\} \leq c(\alpha, N, s) \sup \left\{n^{\alpha}E_n(f):\; n \geq N \right\},$$
is valid, where, depending on the triple $(α,N,s)$ the number $N^{∗}$ may depend on $α,N,Y_s$, and $f$ or be independent of these parameters. |
| first_indexed | 2026-03-24T02:31:59Z |
| format | Article |
| fulltext |
UDC 517.5
K. Kopotun* (Univ. Manitoba, Winnipeg, Canada),
D. Leviatan (Tel Aviv Univ., Israel),
I. A. Shevchuk (Nat. Taras Shevchenko Univ. Kyiv, Ukraine)
ARE THE DEGREES OF BEST (CO)CONVEX
AND UNCONSTRAINED POLYNOMIAL APPROXIMATION
THE SAME? II
ЧИ ОДНАКОВI ПОРЯДКИ НАЙКРАЩОГО (КО)ОПУКЛОГО
НАБЛИЖЕННЯ ТА ПОЛIНОМIАЛЬНОГО НАБЛИЖЕННЯ
БЕЗ ОБМЕЖЕНЬ? II
In Part I of this paper, we proved that for every α > 0 and a continuous function f , which is either convex
(s = 0) or changes convexity at a finite collection Ys = {yi}si=1 of points yi ∈ (−1, 1), one has
sup{nαE(2)
n (f, Ys) : n ≥ N ∗} ≤ c(α, s) sup{nαEn(f) : n ≥ 1},
where En(f) and E(2)
n (f, Ys) denote, respectively, the degrees of best unconstrained and (co)convex approxi-
mation, and c(α, s) is a constant depending only on α and s. Moreover, we showed that N ∗ may be chosen
to be 1 if s = 0 or s = 1, α 6= 4, and that it has to depend on Ys and α if s = 1, α = 4 or s ≥ 2.
In this Part II, we show that a more general inequality
sup{nαE(2)
n (f, Ys) : n ≥ N ∗} ≤ c(α,N , s) sup{nαEn(f) : n ≥ N},
is valid, where, depending on the triple (α,N , s), N ∗ may or may not depend on α, N , Ys and f.
У частинi I цiєї статтi доведено, що для кожного α > 0 та неперервної функцiї f , яка або опукла
(s = 0) або змiнює опуклiсть у скiнченному наборi Ys = {yi}si=1 точок yi ∈ (−1, 1),
sup{nαE(2)
n (f, Ys) : n ≥ N ∗} ≤ c(α, s) sup{nαEn(f) : n ≥ 1},
де En(f) та E
(2)
n (f, Ys) означають вiдповiдно порядок найкращого наближення без обмежень та
(ко)опуклого наближення, c(α, s) є сталою, що залежить лише вiд α i s. Бiльш того, було показа-
но, що N ∗ можна вибрати рiвним одиницi, якщо s = 0 або s = 1, α 6= 4, i що воно повинно залежати
вiд Ys i α, якщо s = 1, α = 4 або s ≥ 2.
У частинi II показано, що виконується бiльш загальна нерiвнiсть
sup{nαE(2)
n (f, Ys) : n ≥ N ∗} ≤ c(α,N , s) sup{nαEn(f) : n ≥ N},
де в залежностi вiд трiйки (α,N , s) число N ∗ може залежати або нi вiд α, N , Ys та f.
1. Introduction and main results. Let C[−1, 1] be the space of continuous functions on
[−1, 1] equipped with the uniform norm ‖·‖, and let Ys, s ∈ N, be the set of all collections
Ys :=
{
yi
}s
i=1
of points yi, such that ys+1 := −1 < ys < . . . < y1 < 1 =: y0. For
Ys ∈ Ys denote by ∆2(Ys) the set of all piecewise convex functions f ∈ C[−1, 1], that
change convexity at the points Ys, and are convex on [y1, 1]. In particular, Y0 = {∅},
and ∆2 = ∆2(Y0) denotes the set of all convex continuous functions. If f is twice
continuously differentiable in (−1, 1), then f ∈ ∆2(Ys) if and only if f ′′(x)Π(x;Ys) ≥
≥ 0, x ∈ (−1, 1), where Π(x;Ys) :=
∏s
i=1
(x− yi), (Π(x, Y0) :≡ 1).
We also denote by
En(f) := inf
{
‖f − Pn‖ : Pn ∈ Pn
}
*Supported by NSERC of Canada.
c© K. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK, 2010
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3 369
370 K. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
and
E(2)
n (f, Ys) := inf
{
‖f − Pn‖ : Pn ∈ Pn ∩∆2(Ys)
}
the degrees of best unconstrained and coconvex approximation of a function f by
polynomials from Pn, the space of algebraic polynomials of degree < n. In particular,
E(2)
n (f) := E(2)
n (f, Y0) = inf
{
‖f − Pn‖ : Pn ∈ Pn ∩∆2
}
is the degree of best convex approximation of f.
While it is obvious that En(f) ≤ E
(2)
n (f), Lorentz and Zeller [1] showed that the
inverse inequality E
(2)
n (f) ≤ cEn(f), is invalid even if a constant c is allowed to
depend on the function f ∈ ∆2. There are many examples showing that the same is
true for piecewise convex functions from ∆2(Ys). The existence of counterexamples
notwithstanding, we recently have proved the following result.
Theorem A [2]. For each α > 0 and integer s ≥ 0 there is a constant c(α, s), such
that for every collection Ys ∈ Ys and a function f ∈ ∆2(Ys) we have
sup
{
nαE(2)
n (f, Ys) : n ≥ N ∗
}
≤ c(α, s) sup
{
nαEn(f) : n ≥ 1
}
, (1.1)
where N ∗ = 1, if either s = 0, or s = 1 and α 6= 4, and N ∗ = N ∗(α, Ys) — a constant,
depending only on α and Ys, if either s ≥ 2, or s = 1 and α = 4.
We also have shown that Theorem A cannot be improved, that is, if either s ≥ 2, or
s = 1 and α = 4, then the constant N ∗ cannot be made independent of Ys.
Theorem B [2]. Let s ≥ 2. Then for every α > 0 and m ∈ N, there exist a
collection Ys ∈ Ys and a function f ∈ ∆2(Ys), such that
mαE(2)
m (f, Ys) ≥ c(α, s)mα+1−dαe sup
{
nαEn(f) : n ≥ 1
}
, (1.2)
where c(α, s) is a positive constant and dαe is the ceiling function (i.e., the smallest
integer not less than α).
Theorem C [2]. For every Y1 ∈ Y1 there exists a function f ∈ ∆2(Y1), satisfying
sup
{
n4En(f) : n ∈ N
}
= 1,
such that for each m ∈ N, we have
m4E(2)
m (f, Y1) ≥
(
c ln
m
1 +m2ϕ(y1)
− 1
)
, (1.3)
and
sup
{
n4E(2)
n (f, Y1) : n ∈ N
}
≥ c |lnϕ(y1)|, (1.4)
where ϕ(y) :=
√
1− y2 and c is an absolute positive constant.
Everywhere below, we denote by c(. . . ) positive real constants that depend only
on the parameters, sets, functions in the parentheses and which may vary from one
occurrence to another even when they appear in the same line. In particular, c denote
absolute positive constants. Similarly, N (. . . ) denote natural numbers that depend only
on the quantities in the parentheses. For instance, N (α, Ys) denotes a natural number
that depends only on α and Ys and nothing else.
The main goal in this paper is to answer the following questions:
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3
ARE THE DEGREES OF BEST (CO)CONVEX AND UNCONSTRAINED POLYNOMIAL . . . 371
What happens if we replace n ≥ 1 in (1.1) by n ≥ N , where N ∈ N? Is
Theorem A still valid? What can be said about the dependence of N ∗ on α,
N , Ys and f?
Our first result is the following generalization of Theorem A.
Theorem 1.1. For each α > 0, N ∈ N, s ∈ N0 := N ∪ {0}, Ys ∈ Ys and
f ∈ ∆2(Ys), there exists an N ∗ ∈ N, such that
sup
{
nαE(2)
n (f, Ys) : n ≥ N ∗
}
≤ c(α,N , s) sup
{
nαEn(f) : n ≥ N
}
. (1.5)
Note that N ∗ ∈ N in the statement of Theorem 1.1 may or may not depend on α,
N , Ys and f. Our Theorem 1.2 below provides a complete answer to when and how this
dependence occurs.
It is rather easy to see that the assertion of Theorem 1.1 in the case N = 2 immedi-
ately follows from Theorem A. Namely,
if N = 2, then Theorem 1.1 is valid with N ∗ = 2, if either s = 0, or s = 1
and α 6= 4, and N ∗ = N ∗(α, Ys) if either s ≥ 2, or s = 1 and α = 4.
Indeed, noting that the function g := f−p2,where p2 := arg infp∈P2 ‖f−p‖, satisfies
En(g) = En(f), E(2)
n (g, Ys) = E
(2)
n (f, Ys) for all n ≥ 2, and E1(g) ≤ ‖g‖ = E2(f),
we have
sup
{
nαE(2)
n (f, Ys) : n ≥ N ∗
}
= sup
{
nαE(2)
n (g, Ys) : n ≥ N ∗
}
≤
≤ c(α, s) sup
{
nαEn(g) : n ≥ 1
}
= c(α, s) sup
{
nαEn(f) : n ≥ 2
}
.
Moreover, Theorems B and C imply that
if N ∗ = 2, then N ∗ cannot be made independent of Ys if either s ≥ 2, or
s = 1 and α = 4.
We now emphasize that, except when 3 ≤ N ≤ s+2, N ∗ cannot be smaller than N .
Indeed, to see this it suffices to consider any function fs ∈ ∆2(Ys) which is a polynomial
of degree exactly N − 1, for instance, such that f ′′s (x) := (x + 2)N−s−3Π(x;Ys) if
N ≥ s + 3, and fs(x) := x if N = 2. Then, En(fs) = 0 for all n ≥ N , and one
immediately gets a contradiction assuming that N ∗ in (1.5) is strictly smaller than N . If
3 ≤ N ≤ s+ 2, then PN ∩∆2(Ys) = P2 ∩∆2(Ys) (any polynomial of degree ≤ s+ 1
which has s convexity changes must be linear), and so E
(2)
N (f, Ys) = E
(2)
2 (f, Ys) =
= E2(f), i.e., if (1.5) is valid with N ∗ = N , then it is also valid with N ∗ = 2.
Also, by Theorem B one may not expect, for s ≥ 2, that N ∗ be independent of Ys.
Given a triple (α,N , s), we want to determine the exact dependence of N ∗ on all
the quantities appearing in the statement of Theorem 1.1 so that (1.5) is satisfied.
We will show that there are three different types of behavior of N ∗, and in order to
describe them we introduce the following notations.
Definition. Let (α,N , s) ∈ R+ × N× N0.
1. We write (α,N , s) ∈ “+”, if Theorem 1.1 holds with N ∗ = N .
2. We write (α,N , s) ∈ “⊕”, if
(a) Theorem 1.1 holds with N ∗ = N ∗(α,N , Ys), and
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3
372 K. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
(b) Theorem 1.1 is not valid with N ∗ which is independent of Ys, that is, for each
A > 0 and M ∈ N there are a number m > M, a collection Ys ∈ Ys, and a function
f ∈ ∆2(Ys), such that
mαE(2)
m (f, Ys) ≥ A sup
{
nαEn(f) : n ≥ N
}
. (1.6)
3. We write (α,N , s) ∈ “ ”, if
(a) Theorem 1.1 holds with N ∗ = N ∗(α,N , Ys, f), and
(b) Theorem 1.1 is not valid with N ∗ which is independent of f, that is, for each
A > 0, M ∈ N, and Ys ∈ Ys, there are m > M and f ∈ ∆2(Ys), such that (1.6) holds.
It turns out that N ∗ depends on
α := dα/2e (1.7)
rather than on α itself with the only exception in the case α = 2, N ≤ 2 and s = 1,
which has already been discussed above.
Theorem 1.2. Let (α,N , s) ∈ R+ × N× N0. Then
(i) (α,N , s) ∈ “ + ”] if
s = 0, α ≤ 2 and N ≤ 3;
s = 0, α ≥ 3 and N ∈ N;
s = 1, α = 1 and N ≤ 2;
s = 1, α = 2, α 6= 4 and N ≤ 2;
s = 1, α = 3 and N ≤ 4;
s = 1, α ≥ 4 and N ∈ N.
(ii) (α,N , s) ∈ “ ”] if
s ≥ 0, α ≤ 2 and N ≥ s+ 4;
s ≥ 1, α = 1 and N = s+ 3.
(iii) (α,N , s) ∈ “ ⊕ ” in all other cases, except perhaps the case s ≥ 3, α = 2 and
N = s+ 3.
We recall that the casesN = 1 andN = 2 in this theorem follow from Theorems A –
C and the discussion following the statement of Theorem 1.1.
In order to make it easier to see and remember what Theorem 1.2 establishes, and to
recognize the patterns of behavior of the triples (α,N , s), we summarize the results in
tables relating N and α, for the various values of s.
The symbol “
◦
+” in the positions (α,N ) = (2, 1) and (2, 2) for s = 1 (the exceptional
case) means that (α,N , s) ∈ “+” if α 6= 4 (i.e., 2 < α < 4), and (α,N , s) ∈ “⊕” if
α = 4.
We also write “?” in the position (α,N ) = (2, s+3) for s ≥ 3 since we do not know
exactly what happens in this case. We do know, however, that (α,N , s) ∈ “ ” or “⊕”,
when s ≥ 3, 2 < α ≤ 4 and N = s+ 3 (see Theorem B and case 11 in Section 4.2).
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3
ARE THE DEGREES OF BEST (CO)CONVEX AND UNCONSTRAINED POLYNOMIAL . . . 373
α
...
...
...
...
... . .
.
4 + + + + + · · ·
3 + + + + + · · ·
2 + + + · · ·
1 + + + · · ·
1 2 3 4 5 N
s = 0
α
...
...
...
...
...
... . .
.
5 + + + + + + · · ·
4 + + + + + + · · ·
3 + + + + ⊕ ⊕ · · ·
2
◦
+
◦
+ ⊕ ⊕ · · ·
1 + + ⊕ · · ·
1 2 3 4 5 6 N
s = 1
α
...
...
...
...
...
...
... . .
.
4 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ · · ·
3 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ · · ·
2 ⊕ ⊕ ⊕ ⊕ ⊕ · · ·
1 ⊕ ⊕ ⊕ ⊕ · · ·
1 2 3 4 5 6 7 N
s = 2
α
...
...
...
...
...
...
...
... . .
.
4 ⊕ ⊕ · · · ⊕ ⊕ ⊕ ⊕ ⊕ · · ·
3 ⊕ ⊕ · · · ⊕ ⊕ ⊕ ⊕ ⊕ · · ·
2 ⊕ ⊕ · · · ⊕ ⊕ ? · · ·
1 ⊕ ⊕ · · · ⊕ ⊕ · · ·
1 2 · · · s+ 1 s+ 2 s+ 3 s+ 4 s+ 5 N
s ≥ 3
2. Proofs of the negative results. We first state the following well known result
(see, e.g., [3, p. 418], Theorem 7.5.2).
Lemma 2.1. Let r ∈ N and Gr(x) = (x+ 1)r ln(x+ 1), Gr(−1) := 0. Then
En(Gr) ≤ c(r)n−2r, n ∈ N. (2.1)
Next we prove the following lemma.
Lemma 2.2. For every A > 0 and m ∈ N, there are points y1 ∈ (−1, 1) and
ỹ1 ∈ (−1, 1), and functions f ∈ ∆2(Y1) and f̃ ∈ ∆2(Ỹ1), where Y1 := {y1} and
Ỹ1 := {ỹ1}, such that
n4En(f) ≤ 1, n ≥ 3, and n6En(f̃) ≤ 1, n ≥ 5, (2.2)
while
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3
374 K. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
E(2)
m (f, Y1) ≥ A and E(2)
m (f̃ , Ỹ1) ≥ A.
Proof. Given A > 0 and m ∈ N, in the proof of [4] (Theorem 2.4), we have
constructed functions g4 ∈ ∆2(Y1) and g6 ∈ ∆2(Ỹ1), for some −1 < y1 < 1 and
−1 < ỹ1 < 1, such that
E(2)
m (g4, Y1) ≥ A and E(2)
m (g6, Ỹ1) ≥ A. (2.3)
The functions had the representation g2r = P2r−1 + crGr, r = 2, 3, where P2r−1 ∈
∈ P2r−1 and cr is an absolute constant. By virtue of (2.1) we therefore conclude that
n2rEn(g2r) ≤ c, n ≥ 2r − 1,
and the proof is complete.
Remark 2.1. Note that Lemma 2.2 readily implies that if s = 1, then for α = 1, 2
and all N ≥ 3 as well as for α = 3 and all N ≥ 5, there cannot be “+” in the position
(α,N ).
Our next result is valid for arbitrary s ∈ N0.
Lemma 2.3. Let s ∈ N0 and Ys ∈ Ys. For every A > 0 and m ∈ N, there is a
function f ∈ ∆2(Ys), such that
n4En(f) ≤ 1, n ≥ s+ 4,
while
E(2)
m (f, Ys) ≥ A.
Proof. Following [4], for each b ∈ (−1, 0), we denote
fb(x) :=
x∫
0
(x− t)Π(t;Ys)
t∫
b
t− u
(u+ 1)2
du
dt.
Clearly, f ′′b (x)Π(x;Ys) ≥ 0, x ∈ (−1, 1), so that fb ∈ ∆2(Ys). Straightforward
computations using the Taylor expansion of Π(x;Ys) about t = −1, yield,
fb = Ps+4 −
s∑
r=0
Π(r)(−1;Ys)
(r + 2)!
Gr+2,
where Ps+4 ∈ Ps+4. Hence, by virtue of Lemma 2.1, we obtain,
n4En(fb) ≤ c(s), n ≥ s+ 4, (2.4)
since
∥∥Π(r)(·;Ys)
∥∥ ≤ c(s), 0 ≤ r ≤ s.
The polynomial
ps+4(x) :=
x∫
0
(x− t)Π(t;Ys)
1∫
b
t− u
(u+ 1)2
du
dt,
belongs to Ps+4 and satisfies Π(−1;Ys)p′′s+4(−1) = Π2(−1;Ys) ln
b+ 1
2
. Hence, for
each polynomial Pm ∈ Pm ∩∆2(Ys), m ≥ s+ 4, we have
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3
ARE THE DEGREES OF BEST (CO)CONVEX AND UNCONSTRAINED POLYNOMIAL . . . 375
−Π2(−1;Ys) ln
b+ 1
2
= −Π(−1;Ys)p′′s+4(−1) ≤
≤ Π(−1;Ys)
(
P ′′m(−1)− p′′s+4(−1)
)
≤
≤ m4|Π(−1;Ys)|‖Pm − ps+4‖, (2.5)
where we used Markov’s inequality. Also
ps+4(x)− fb(x) =
x∫
0
(x− t)Π(t;Ys)
1∫
t
t− u
(u+ 1)2
du
dt,
which is independent of b. Hence, by (2.5),
m−4
∣∣Π(−1;Ys)
∣∣ ln 2
b+ 1
≤ ‖Pm − fb‖+ ‖fb − ps+4‖ ≤ ‖Pm − fb‖+ c(s).
Thus,
E(2)
m (fb, Ys) ≥ m−4
∣∣Π(−1;Ys)
∣∣ ln 2
b+ 1
− c(s),
and taking f := cfb with suitable c = c(s) and b concludes the proof of the lemma.
Remark 2.2. Lemma 2.3 implies that if α = 1 or 2, then for all s ≥ 0 and
N ≥ s+ 4, there cannot be “+” or “⊕” in the position (α,N ) (and so the best we can
hope for is that there is “ ” in those positions which, as will be shown below, is indeed
the case).
Finally, for s ≥ 1, we have the following lemma.
Lemma 2.4. Let s ∈ N and Ys ∈ Ys. For each A > 0 and m ∈ N, there is a
function f ∈ ∆2(Ys), such that
n2En(f) ≤ 1, n ≥ s+ 3,
and
E(2)
m (f, Ys) ≥ A.
Proof. Denote Dj(x) := xj ln |x| (Dj(0) := 0). It is well known (and is easy to
check) that for j ≥ 1, D(j−1)
j belongs to the Zygmund class, i.e., ω2(D(j−1)
j , t) ≤ c(j)t.
Thus, for j ≥ 2, En(Dj) ≤ c(j)n−j ≤ c(j)n−2, n ≥ 1. Hence, for Dj,γ(x) :=
:= Dj(x+ γ), −1 < γ < 1, j ≥ 2, it follows that
En(Dj,γ) ≤ c(j)
n2
n ≥ 1. (2.6)
Take 0 < b <
1
2
min{y1 − y2, 1− y1}, and let
l̃b(x) :=
x
b
− 1 + ln b. (2.7)
(Note that y = lb(x) is the tangent to the function ln |x| at the point x = b.) Further, let
b∗ be the other (clearly negative) root of the equation l̃b(x) = ln |x|. Clearly,
|b∗| = −b∗ < b, (2.8)
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376 K. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
and (x− b∗)
(
l̃b(x)− ln |x|
)
≥ 0, x 6= 0, so that for
lb(x) := l̃b(x+ b∗), (2.9)
we have
x(lb(x)− ln |x+ b∗|) ≥ 0, x 6= |b∗|. (2.10)
Denote
Π1(x) := Πs
i=2(x− yi)
(Π1 :≡ 1 if s = 1), and let
Lb(x) :=
x∫
0
(x− u)Π1(u)lb(u− y1)du,
and
gb(x) :=
x∫
0
(x− u)Π1(u) ln |u+ b∗ − y1|du.
Finally, write
fb := Lb − gb.
Integration by parts yields
x∫
0
(x− u) ln |u+ b∗ − y1|du =
1
2
D2(x+ b∗ − y1) + p3(x),
where p3 ∈ P3. Similarly,
gb(x) =
s−1∑
r=0
Π(r)
1 (y1 − b∗)
(r + 2)!
Dr+2(x+ b∗ − y1) + ps+2(x), (2.11)
where ps+2 ∈ Ps+2, and since Lb ∈ Ps+3, (2.6) yields
En(fb) ≤
c(s)
n2
, n ≥ s+ 3. (2.12)
At the same time, it follows by (2.10) that fb ∈ ∆2(Ys).
On the other hand, given Pm ∈ Pm ∩∆2(Ys), we conclude by (2.7) through (2.9)
that
0 < Π1(y1) ln
1
b
< Π1(y1)
(
ln
1
b
+ 1− b∗
b
)
=
= −L′′b (y1) = P ′′m(y1)− L′′b (y1) ≤ c(s, y1)m2‖Pm − Lb‖,
where we applied Bernstein’s inequality. Since
‖gb‖ ≤ 2‖Π1‖
1∫
0
| lnx|dx = 2‖Π1‖ ≤ 2s,
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then
0 < Π1(y1) ln
1
b
≤ c(s, y1)m2
(
‖Pm − fb‖+ ‖gb‖
)
≤ c(Ys)m2(‖Pm − fb‖+ 1).
Hence
E(2)
m (fb, Ys) ≥
c(Ys)
m2
ln
1
b
− 1,
and this combined with (2.12) implies the statement of the lemma for f := cfb with
suitable c = c(s) and b.
Remark 2.3. Lemma 2.4 implies that if α = 1 and s ≥ 1, then for all N ≥ s+ 3,
there cannot be “+” or “⊕” in the position (α,N ) (and so the best we can hope for is
that there is “ ” in those positions which is indeed the case, see below).
3. Auxiliary results. Recall that ϕ(x) =
√
1− x2, and let Crϕ, r ≥ 1, be the space
of functions f ∈ Cr(−1, 1) ∩ C[−1, 1] such that
lim
x→±1
ϕr(x)f (r)(x) = 0,
and C0
ϕ := C[−1, 1].
If
∆k
δ (g, x) :=
k∑
i=0
(
k
i
)
(−1)k−ig
(
x− kδ
2
+ iδ
)
,
denotes the k-th symmetric difference of a function g with a step δ, then the Ditzian –
Totik type modulus of smoothness of the rth derivative of a function f ∈ Crϕ, is
defined by
ωϕk,r(f
(r), t) := sup
h∈[0,t]
sup
x:|x|+(kh)ϕ(x)/2<1
W r
(
x,
kh
2
)∣∣∣∆k
hϕ(x)(f
(r), x)
∣∣∣ , (3.1)
with the weight
W (x, µ) := ϕ
(
|x|+ µϕ(x)
)
, |x|+ µϕ(x) < 1. (3.2)
If r = 0, then
ωϕk (f, t) := ωϕk,0(f, t)
is the (usual) Ditzian – Totik modulus of smoothness. Finally, let ‖f‖C[a,b] denote the
uniform norm of a function f ∈ C[a, b] (in particular, ‖f‖C[−1,1] = ‖f‖) and recall that
the ordinary k-th modulus of smoothness of f ∈ C[a, b] is
ωk
(
f, t, [a, b]
)
:= sup
h∈[0,t]
∥∥∆k
h(f, ·)
∥∥
C[a+kh/2,b−kh/2],
and denote ωk(f, t) := ωk(f, t, [−1, 1]).
The following results are so-called inverse theorems. They characterize the smoothness
(i.e., describe the class) of functions that have the prescribed order of polynomial approxi-
mation.
First we formulate a corollary of the classical Dzyadyk – Timan – Lebed – Brudnyi
inverse theorem (see, e.g., [3], Theorem 7.1.2).
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378 K. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
Theorem 3.1. Let 2r < α < 2k + 2r, and f ∈ C[−1, 1]. If
nαEn(f) ≤ 1, n ≥ k + r,
then f ∈ Cr[−1, 1] and
ωk(f (r), t2) ≤ c(α, k, r)tα−2r. (3.3)
For the Ditzian – Totik type moduli of smoothness we need the following result which
is a generalization of [5] (Theorem 7.2.4) in the case p =∞.
Denote by Φ the set of nondecreasing functions φ : [0,∞) → [0,∞), satisfying
φ(0+) = 0.
Theorem 3.2. Given k ∈ N, r ∈ N0, N ∈ N, and φ ∈ Φ such that
1∫
0
rφ(u)
ur+1
du < +∞.
If
En(f) ≤ φ
(
1
n
)
, for all n ≥ N,
then f ∈ Crϕ, and
ωϕk,r(f
(r), t) ≤ c(k, r)
t∫
0
rφ(u)
ur+1
du+ c(k, r)tk
1∫
t
φ(u)
uk+r+1
du+
+c(k, r,N)tkEk+r(f), t ∈ [0, 1/2].
If, in addition, N ≤ k + r, then the following Bari – Stechkin type estimate holds:
ωϕk,r(f
(r), t) ≤ c(k, r)
t∫
0
rφ(u)
ur+1
du+ c(k, r)tk
1∫
t
φ(u)
uk+r+1
du, t ∈ [0, 1/2].
For readers’ sake, we provide a proof of this theorem in the appendix.
In fact, we only need the following theorem which is an immediate consequence of
Theorem 3.2 (φ(u) := uα), but is of special interest in the context of this paper.
Theorem 3.3. Let r ∈ N0, k ∈ N and α > 0, be such that r < α < k+ r, and let
f ∈ C[−1, 1]. If
nαEn(f) ≤ 1, for all n ≥ N,
where N ≥ k + r, then f ∈ Crϕ and
ωϕk,r(f
(r), t) ≤ c(α, k, r)tα−r + c(N, k, r)tkEk+r(f).
In particular, if N = k + r, then
ωϕk,r(f
(r), t) ≤ c(α, k, r)tα−r.
Lemma 3.1 ([6], [4] (Theorems 2.7, 2.8 and 2.11), [2] (Lemma 2.8), [7] (Theo-
rem 3.1)).
I. Let f ∈ ∆2. If f ∈ C[−1, 1], then
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E(2)
n (f) ≤ cωϕ4
(
f,
1
n
)
+ cn−6‖f‖, n ≥ 3. (3.4)
Moreover, if f ∈ C2
ϕ ∩ C1[−1, 1], then
E(2)
n (f) ≤ c(k)n−2ωϕk,2
(
f ′′,
1
n
)
+ c(k)n−2ω2
(
f ′,
1
n2
)
, n ≥ 3. (3.5)
Furthermore, if f ∈ C2
ϕ ∩ C2[−1, 1], and k, l ∈ N, then, for n ≥ l + 2, we have
E(2)
n (f) ≤ c(k, l)n−2ωϕk,2
(
f ′′,
1
n
)
+ c(k, l)n−4ωl
(
f ′′,
1
n2
)
. (3.6)
II. Let f ∈ ∆2(Y1). If f ∈ C[−1, 1], then
E(2)
n (f, Y1) ≤ cωϕ3
(
f,
1
n
)
+ cω2
(
f,
1
n2
)
, n ≥ 2. (3.7)
If, in addition, f ∈ C2
ϕ ∩ C1[−1, 1], then
E(2)
n (f, Y1) ≤ cn−2ωϕ3,2
(
f ′′,
1
n
)
+ cn−2ω1
(
f ′,
1
n2
)
, n ≥ 2, (3.8)
and
E(2)
n (f, Y1) ≤ cn−2ωϕ3,2
(
f ′′,
1
n
)
+ cn−2ω2
(
f ′,
1
n2
)
, nϕ(y1) > 1. (3.9)
If f ∈ C2
ϕ, then
E(2)
n (f, Y1) ≤ cn−2ωϕ3,2
(
f ′′,
1
n
)
+ cn−4ωϕ2,2
(
f ′′,
1
n
)
, n ≥ N(Y1), (3.10)
and
E(2)
n (f, Y1) ≤ cn−2ωϕ3,2
(
f ′′,
1
n
)
, n ≥ N(f). (3.11)
Moreover, if we actually have f ∈ C3
ϕ ∩ C2[−1, 1], then for any k ∈ N,
E(2)
n (f, Y1) ≤ c(k)n−3ωϕk,3
(
f (3),
1
n
)
+ c(k)n−4ω2
(
f ′′,
1
n2
)
, n ≥ 4. (3.12)
Furthermore, if f ∈ C3[−1, 1], then
E(2)
n (f, Y1) ≤ c(k)n−3ωϕk,3
(
f (3),
1
n
)
+c(k)n−6ωk
(
f (3),
1
n2
)
, n ≥ k+3. (3.13)
III. Let f ∈ ∆2(Ys), s ∈ N. If f ∈ C[−1, 1], then
E(2)
n (f, Ys) ≤ c(s)ωϕ3
(
f,
1
n
)
, n ≥ N(Ys). (3.14)
Moreover, if f ∈ C3
ϕ∩C2[−1, 1], s ∈ N, and k, l ∈ N, then there exists N(Ys, k, l) such
that, for all n ≥ N(Ys, k, l),
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380 K. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
E(2)
n (f, Ys) ≤ c(k, l, s)n−3ωϕk,3
(
f (3),
1
n
)
+ c(k, l, s)n−4ωl
(
f ′′,
1
n2
)
. (3.15)
Also, if s ≥ 2 and f ∈ C2
ϕ, then
E(2)
n (f, Ys) ≤ c(s)n−2ωϕ3,2
(
f ′′,
1
n
)
, n ≥ N(Ys). (3.16)
Remark 3.1. Estimate (3.13) was not proved in [2]. However, its proof is very
similar to those in [2], and it is based upon the fact that if f ∈ C3[a, b] is such that f is
concave on [a, y1] and convex on [y1, b] (i.e., f ′′(x)(x− y1) ≥ 0, a ≤ x ≤ b), and pk is
such that pk ≥ f (3) on [a, b] and ‖f (3) − pk‖ ≤ c(k)ωk(f (3), b− a, [a, b]) (for example,
pk := arg infp∈Pk
‖f (3) − p‖C[a,b] + infp∈Pk
‖f (3) − p‖C[a,b]), then
P (x) :=
x∫
a
t∫
a
s∫
y1
pk(v) dv ds dt+ f(a) + f ′(a)(x− a)
is a polynomial from Pk+3 that is coconvex with f on [a, b] and satisfies P (a) = f(a)
and
‖f − P‖C[a,b] ≤ c(k)(b− a)3ωk
(
f (3), b− a, [a, b]
)
. (3.17)
We omit the details.
4. Proofs of the positive results. Since the cases N = 1 and N = 2 have already
been discussed we assume thatN ≥ 3. Given α > 0, integersN ≥ 3, s ≥ 0, a collection
Ys ∈ Ys, and a function f ∈ ∆2(Ys), assume without loss of generality that
nαEn(f) ≤ 1, for all n ≥ N . (4.1)
Then we have to prove the inequality
nαE(2)
n (f, Ys) ≤ c(α,N , s), n ≥ N ∗, (4.2)
with a proper N ∗.
4.1. Convex approximation: s = 0.
1. N = 3, 0 < α < 3 (“+”).
Theorem 3.3 (with r = 0 and k = 3), inequality (4.1), and the estimate E(2)
n (f) ≤
≤ cωϕ3 (f, 1/n), n ≥ 3, proved in [8], yield E
(2)
n (f) ≤ cωϕ3 (f, 1/n) ≤ cn−α, for
n ≥ 3 =: N ∗.
2. N = 3, 3 ≤ α ≤ 4 (“+”).
Theorem 3.3 (with r = 2 and k = 3), Theorem 3.1 (with r = 1 and k = 2),
and inequality (4.1), imply that f ∈ C2
ϕ ∩ C1[−1, 1], ωϕ3,2(f ′′, t) ≤ c(α)tα−2, and
ω2(f ′, t2) ≤ c(α)tα−2. Inequality (3.5) now yields E(2)
n (f) ≤ c(α)n−α for n ≥ 3 =:
=: N ∗.
3. α > 4, N > α (“+”).
Theorem 3.3 (with r = 2 and k = N−2), Theorem 3.1 (with r = 2 and k = N−2),
and inequality (4.1), imply that f ∈ C2
ϕ∩C2[−1, 1], ωϕN−2,2(f ′′, t) ≤ c(α,N )tα−2, and
ωN−2(f ′′, t2) ≤ c(α,N )tα−4. Therefore, (3.6) (with k = l = N − 2) yields (4.2) with
N ∗ = N .
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4. α > 4, 4 < N ≤ α (“+”).
Let N1 := bαc + 1 and note that N1 > α ≥ N . Since (4.1) is satisfied with
N1 instead of N , it follows from case 3 that nαE(2)
n (f) ≤ c(α), n ≥ N1. Now, let
α1 := N/2+2 and note that 4 < α1 < N . It follows from (4.1) that nα1En(f) ≤ 1, for
all n ≥ N , and using case 3 again we get nα1E
(2)
n (f) ≤ c(N ), n ≥ N . Therefore, for
N ≤ n < N1, we have nαE(2)
n (f) ≤ c(N )nα−α1 ≤ c(N )Nα−α1
1 ≤ c(α,N ), which
verifies (4.2) with N ∗ = N .
5. N = 3, α > 4 (“+”).
It follows from cases 3 and 4 that (4.2) is valid for n ≥ 5. We note that the
polynomial of best approximation of degree ≤ 2 to a convex function f has to be
convex (this follows, for example, from the Chebyshev Equioscillation Theorem), and
so E(2)
3 (f) = E3(f). Hence, for n = 3 and 4, we have
E(2)
n (f) ≤ E(2)
3 (f) = E3(f) ≤ 1 ≤ 4αn−α,
and so (4.2) with N ∗ = 3 follows.
6. N = 4, α > 4 (“+”).
As in case 5, it follows from cases 3 and 4 that (4.2) is valid for n ≥ 5 and so
we only need to show that E(2)
4 (f) ≤ c(α). Since (4.1) implies that nα1En(f) ≤ 1,
n ≥ 4, where α1 := min{α, 5}, it follows from Theorem 3.1 (with r = 2 and k = 2)
that f ∈ C2[−1, 1] and ω2(f ′′, t2) ≤ c(α)tα1−4 and, in particular, ω2(f ′′, 1) ≤ c(α).
Therefore, E2(f ′′) ≤ cω2(f ′′, 1) ≤ c(α). Now, since the inequality E(2)
4 (f) ≤ 2E2(f ′′)
holds for each f ∈ C2[−1, 1] ∩∆2, we conclude that E(2)
4 (f) ≤ c(α) as needed.
7. N ≥ 4, 0 < α < 4 (“ ”).
Theorem 3.3 (with k = 4 and N = N ), and inequalities (4.1) and (3.4), yield
E(2)
n (f) ≤ c(α)n−α + c(N )n−4‖f‖ ≤ c(α)n−α,
for all n ≥ max
{
3, c(α,N )‖f‖1/(4−α)
}
=: N ∗.
8. N ≥ 4, α = 4 (“ ”).
Theorem 3.3 (with r = 2 and k = 3), Theorem 3.3 (with r = 1 and k = 3), and
inequality (4.1), imply that f ∈ C2
ϕ∩C1[−1, 1], ωϕ3,2(f ′′, t) ≤ ct2, and ω3(f ′, t2) ≤ ct2.
By the Marchaud classical inequality (see, e.g., [5], (4.3.1)) the latter estimate implies
ω2(f ′, t) ≤ ct + ct2‖f ′‖. Inequality (3.5) (with k = 3) now yields E(2)
n (f) ≤ cn−4 +
+ cn−6‖f ′‖, n ≥ 3, and hence (4.2) follows with N ∗ := max
{
3, c‖f ′‖1/2
}
.
4.2. Coconvex approximation: the case s ≥ 1. For some cases below we need
the fact (see [9]), that for any f ∈ ∆2(Ys), s ≥ 1,
E2(f) ≤ c(Ys)Es+2(f). (4.3)
Remark 4.1. For the reader’s convenience, we list in each case below, the full
range of α’s for which that particular proof is suitable. Hence, the same triple (α,N , s)
may be covered by more than one case.
1. s = 1, 4 < α < 8, N = 4 (“+”).
Theorem 3.3 (with r = 3 and k = 5), Theorem 3.3 (with r = k = 2), and inequality
(4.1), imply that f ∈ C3
ϕ ∩ C2[−1, 1], ωϕ5,3(f (3), t) ≤ c(α)tα−3, and ω2(f ′′, t2) ≤
≤ c(α)tα−4. Therefore (3.12) (with k = 5), yields (4.2) with N ∗ = 4.
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382 K. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
2. s = 1, 4 < α < 8, N = 3 (“+”).
It follows from case 1 that (4.2) is satisfied for n ≥ 4. Thus, in order to show
that N ∗ = 3 we only need to verify that E(2)
3 (f, Y1) ≤ c(α). Indeed, since (4.1) is
satisfied with α1 := min{α, 5}, it follows from Theorem 3.2 (with r = 2 and k = 1),
that f ∈ C2[−1, 1] (so that f ′′(y1) = 0) and ω1(f ′′, t2) ≤ c(α)tα1−4 and, in particular,
ω1(f ′′, 1) ≤ c(α). Now take p2(x) := f(y1) + f ′(y1)(x− y1), and we have
E
(2)
3 (f, Y1) = E
(2)
2 (f, Y1) = E2(f) ≤ ‖f − p2‖ =
=
∥∥∥∥∥∥
x∫
y1
u∫
y1
(f ′′(s)− f ′′(y1)) ds du
∥∥∥∥∥∥ ≤ cω1(f ′′, 1) ≤ c(α).
3. s = 1, α > 6, N > α (“+”).
Theorems 3.3 and 3.1 (with r = 3 and k = N − 3), and inequality (4.1), imply
that f ∈ C3 and ωϕN−3,3(f (3), t) ≤ c(α,N )tα−3 and ωN−3(f (3), t2) ≤ c(α,N )tα−6.
Estimate (3.13) (with k = N − 3) now yields (4.2) with N ∗ = N .
4. s = 1, α > 6, 6 < N ≤ α (“+”).
Let N1 := bαc + 1 and note that N1 > α ≥ N . Since (4.1) is satisfied with N1
instead of N , it follows from case 3 that nαE(2)
n (f, Y1) ≤ c(α), n ≥ N1. Now, let
α1 := (N + 6)/2 and note that 6 < α1 < N . It follows from (4.1) that nα1En(f) ≤ 1,
for all n ≥ N , and using case 3 again we get nα1E
(2)
n (f, Y1) ≤ c(N ), n ≥ N .
Therefore, for N ≤ n < N1, we have nαE
(2)
n (f, Y1) ≤ c(N )nα−α1 ≤
≤ c(N )Nα−α1
1 ≤ c(α,N ), which verifies (4.2) with N ∗ = N .
5. s = 1, α > 6, N = 3 (“+”).
It follows from cases 3 and 4, that (4.2) is valid with n ≥ 7. Now, since (4.1) is
obviously valid with, say, α = 5, it follows from case 2 that E(2)
3 (f, Y1) ≤ c, and so,
for 3 ≤ n ≤ 6,
nαE(2)
n (f, Y1) ≤ 6αE(2)
3 (f, Y1) ≤ c(α),
and so (4.2) is valid with N ∗ = 3.
6. s = 1, α > 6, N = 4 (“+”).
The proof is completely analogous to the one in case 5 except that the fact that
E
(2)
4 (f, Y1) ≤ c follows from case 1.
7. s = 1, α > 6, N = 6 (“+”).
It follows from cases 3 and 4, that (4.2) is valid with n ≥ 7. Hence, as in case
5, we only need to show that E(2)
6 (f, Y1) ≤ c(α). If α1 := min{α, 7}, it follows from
(4.1) that nα1En(f) ≤ 1, for all n ≥ 6, so that applying Theorem 3.2 (with r = 3 and
k = 3), we conclude that f ∈ C3[−1, 1] and ω3(f (3), t2) ≤ c(α)tα1−6 and, in particular,
ω3(f (3), 1) ≤ c(α). The inequality E(2)
6 (f, Y1) ≤ c(α) now follows from (3.17) (with
k = 3 and [a, b] = [−1, 1]).
8. s = 1, α > 6, N = 5 (“+”).
The argument is exactly the same as in the previous case with the only difference
that k = 2 is used instead of k = 3.
9. s ≥ 1, 0 < α < 3, 3 ≤ N ≤ s+ 2 (“⊕”).
Theorem 3.189 (with k = 3 and f−p3 in place of f where p3 := arg inf
p∈P3
‖f−p‖),
implies ωϕ3 (f, t) ≤ c(α)tα + c(s)t3E3(f). Now, by (4.3) and (4.1),
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E3(f) ≤ E2(f) ≤ c(Ys)Es+2(f) ≤ c(α, Ys).
Therefore, ωϕ3 (f, 1/n) ≤ c(α)n−α + c(α, Ys)n−3 ≤ c(α)n−α, for n ≥ N(α, Ys).
Inequality (4.2) now follows from (3.14).
10. s ≥ 2, 2 < α < 5, 3 ≤ N ≤ s+ 2 (“⊕”).
Theorem 3.3 (with r = 2, k = 3), implies that f ∈ C2
ϕ and ωϕ3,2(f ′′, t) ≤
≤ c(α)tα−2 + c(s)t3E5(f), and by (4.3) and (4.1) we have
E5(f) ≤ E2(f) ≤ c(Ys)Es+2(f) ≤ c(α, Ys).
Hence, ωϕ3,2(f ′′, 1/n) ≤ c(α)n−α+2 + c(α, Ys)n−3 ≤ c(α)n−α+2, for n ≥ N(α, Ys).
Inequality (4.2) now follows from (3.16).
11. s ≥ 1, 2 < α < 5, N ≥ s+ 3 (“ ”) (except all “⊕” cases in these regions).
As in case 10, we can prove that ωϕ3,2(f ′′, 1/n) ≤ c(α)n−α+2 + c(s)n−3‖f‖, so
that ωϕ3,2(f ′′, 1/n) ≤ c(α)n−α+2, for n ≥ N(α, f). Hence, (4.2) with N ∗ = N ∗(α, f)
follows from (3.16) for s ≥ 2, and from (3.11) for s = 1.
12. s = 2, 2 < α < 5, N = 5 (“⊕”).
Theorem 3.3 (with r = 2 and k = 3) and (4.1), imply that f ∈ C2
ϕ and ωϕ3,2(f ′′, t) ≤
≤ c(α)tα−2. Now, (3.16) implies (4.2) with N ∗ = N ∗(α, Ys).
13. s = 1, 4 < α ≤ 6, N ≥ 5 and s ≥ 2, α > 4, N ≥ 3 (“⊕”).
If N1 := max{bαc + 1,N}, then Theorem 3.3 (with r = 3 and k = N1 − 3),
Theorem 3.3 (with r = 2 and k = N1 − 2), and (4.1), imply that f ∈ C3
ϕ ∩ C2[−1, 1],
ωϕN1−3,3(f (3), t) ≤ c(α,N )tα−3 and ωN1−2(f ′′, t2) ≤ c(α,N )tα−4. Therefore, (3.15)
(with k = N1 − 3 and l = N1 − 2), yields (4.2) with N ∗ = N ∗(α,N , Ys).
14. s = 1, 2 < α < 5, N = 3 or 4 (“⊕”).
Theorem 3.3 (with r = 2 and k = 3) and (4.1), imply that f ∈ C2
ϕ and ωϕ3,2(f ′′, t) ≤
c(α)tα−2. Setting α1 := min{α, 3}, Theorem 3.3 (with r = k = 2) implies that
ωϕ2,2(f ′′, t) ≤ c(α)tα1−2. Therefore, it follows from (3.10) that
E(2)
n (f, Y1) ≤ cn−α + cn−α1−2 ≤ cn−α, n ≥ N(Y1),
as required.
15. s ≥ 1, 0 < α < 3, N ≥ s+ 3 (“⊕”).
Theorem 3.3 (with k = 3 and N = N ), and inequalities (4.1) and (3.14), yield
E
(2)
n (f) ≤ c(α)n−α + c(N )n−3‖f‖ ≤ c(α)n−α, for all sufficiently large n, n ≥
N ∗(α,N , Ys, f).
5. Appendix: proof of Theorem 3.2. We first give the proof for the case r ≥ 1.
Without any loss of generality assume that N ≥ k + r. Set mj := N2j and φj :=
:= φ(m−1
j ). We expand f into the telescopic series
f = Pk+r + (PN − Pk+r) +
∞∑
j=0
(Pmj+1 − Pmj
) =: Pk+r +Q+
∞∑
j=0
Qj , (5.1)
where Pn ∈ Pn are the polynomials of best approximation of f, that is ‖f − Pn‖ =
= En(f). Hence, the polynomialsQj are of degree< mj+1 and satisfy ‖Qj‖ ≤ φj+1+
+ φj ≤ 2φj . For a fixed x ∈ (−1, 1) and h ∈ [0, t], satisfying khϕ(x)/2 < 1− |x|, set
x∗ := |x|+khϕ(x)/2 and note that if u ∈ [−x∗, x∗] ⊇ [x−khϕ(x)/2, x+khϕ(x)/2] =:
=: A, then ϕ(u) ≥ ϕ(x∗). Hence, for u ∈ A and l ∈ N, the Markov – Bernstein
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3
384 K. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
inequality implies,
∣∣Q(l)
j (u)
∣∣ ≤ c(l)ml
j+1
(
1
mj+1
+ ϕ(u)
)−l
φj ≤ c(l)ml
j
(
1
mj
+ ϕ(x∗)
)−l
φj , (5.2)
which in turn yields for l = r,
∣∣∆k
hϕ(x)(Q
(r)
j , x)
∣∣ ≤ 2k max
u∈A
|Q(r)
j (u)| ≤ c(r)2k
mr
j
ϕr(x∗)
φj .
Therefore, if we denote J := min{j : 1/mj ≤ h}, then we have
ϕr(x∗)
∞∑
j=J+1
∣∣∣∆k
hϕ(x)
(
Q
(r)
j , x
)∣∣∣ ≤ c(r)2k ∞∑
j=J+1
mr
jφj =
= c(k, r)
∞∑
j=J+1
m−1
j−1∫
m−1
j
φj
ur+1
du ≤ c(k, r)
∞∑
j=J+1
m−1
j−1∫
m−1
j
φ(u)
ur+1
du =
= c(k, r)
m−1
J∫
0
φ(u)
ur+1
du ≤ c(k, r)
h∫
0
φ(u)
ur+1
du. (5.3)
We also note that
ϕ(x)− ϕ(x∗)
kh/2
=
ϕ(x)− ϕ(x∗)
x∗ − |x|
ϕ(x) <
<
ϕ(x)− ϕ(x∗)
x∗ − |x|
(
ϕ(x) + ϕ(x∗)
)
= x∗ + |x| < 2,
so that
ϕ(x) < kh+ ϕ(x∗).
Hence, for 0 ≤ j ≤ J, taking into account that 1/mj > h/2, we obtain by (5.2) with
l = r + k, ∣∣∆k
hϕ(x)(Q
(r)
j , x)
∣∣ ≤ (hϕ(x))k max
u∈A
∣∣Q(k+r)
j (u)
∣∣ ≤
≤ c(k, r)
hkmk+r
j ϕk(x)
(kh+ ϕ(x∗))k+r
φj ≤ c(k, r)
hkmk+r
j
ϕr(x∗)
φj ≤
≤ c(k, r) hk
ϕr(x∗)
m−1
j−1∫
m−1
j
φj
uk+r+1
du ≤
≤ c(k, r) hk
ϕr(x∗)
m−1
j−1∫
m−1
j
φ(u)
uk+r+1
du,
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3
ARE THE DEGREES OF BEST (CO)CONVEX AND UNCONSTRAINED POLYNOMIAL . . . 385
where m−1 := N/2. Hence, we get
ϕr(x∗)
J∑
j=0
∣∣∣∆k
hϕ(x)
(
Q
(r)
j , x
)∣∣∣ ≤ c(k, r)hk J∑
j=0
m−1
j−1∫
m−1
j
φ(u)
uk+r+1
du =
= c(k, r)hk
2/N∫
m−1
J
φ(u)
uk+r+1
du ≤ c(k, r)hk
1∫
h/2
φ(u)
uk+r+1
du ≤
≤ c(k, r)hk
∫ 1
h
φ(u)
uk+r+1
du, (5.4)
and note that
h∫
0
φ(u)
ur+1
du+ hk
1∫
h
φ(u)
uk+r+1
du ≤
t∫
0
φ(u)
ur+1
du+ tk
1∫
t
φ(u)
uk+r+1
du, h ≤ t.
Finally, we have the estimate∣∣∆k
hϕ(x)(Q
(r), x)
∣∣ ≤ hk∥∥Q(k+r)
∥∥ ≤ 2N2(k+r)hkEk+r(f), (5.5)
which follows by Markov’s inequality. Note that if N = k + r, then Q ≡ 0, so that the
left-hand side of (5.5) vanishes and no estimate is needed
Now, the observation that ∆k
hϕ(x)(P
(r)
k+r, x) = 0, combined with (5.3), (5.4), and
(5.5), completes the proof of the theorem for r ≥ 1.
For r = 0, we write
f = Pk +Q+
J∑
j=0
Qj + (f − PmJ+1),
where Q := PN − Pk and Qj := Pmj+1 − Pmj
(see (5.1)), and complete the proof as
above, just applying (5.4), (5.5) and the inequality
hk
∫ 1
h
φ(u)
uk+1
du ≤ 3tk
1∫
t
φ(u)
uk+1
du, h ≤ t ≤ 1
2
.
Theorem 3.2 is proved.
Remark 5.1. In the definition of the modulus ωϕk,r in this paper, we have used the
weight W (x, µ) from (3.2) where µ = kh/2. Note that we could also use the weights
(see [4, 10])
W1(x, µ) :=
(
(1− µϕ(x))2 − x2
)1/2
,
or (see [2])
W2(x, µ) :=
(
ϕ2(x)− µϕ(x)(1 + |x|)
)1/2
,
which would yield equivalent definitions of the modulus ωϕk,r since, for µ ∈ (0, 1) and
x : |x|+ µϕ(x) < 1, √
1− µ
1 + µ
W ≤W1 ≤W2 ≤W.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3
386 K. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
1. Lorentz G. G., Zeller K. L. Degree of approximation by monotone polynomials. II // J. Approxim.
Theory. – 1969. – 2. – P. 265 – 269.
2. Kopotun K., Leviatan D., Shevchuk I. A. Are the degrees of best (co)convex and unconstrained polynomial
approximation the same? // Acta math. hung. – 2009. – 123, № 3. – P. 273 – 290.
3. Dzyadyk V. K., Shevchuk I. A. Theory of uniform approximation of functions by polynomials. – Berlin:
Walter de Gruyter GmbH & Co. KG, 2008.
4. Kopotun K. A., Leviatan D., Shevchuk I. A. Coconvex approximation in the uniform norm: the final
frontier // Acta math. hung. – 2006. – 110, № 1-2. – P. 117 – 151.
5. Ditzian Z., Totik V. Moduli of smoothness // Springer Ser. Comput. Math. – New York: Springer, 1987.
– Vol. 9.
6. Kopotun K. A., Leviatan D., Shevchuk I. A. The degree of coconvex polynomial approximation // Proc.
Amer. Math. Soc. – 1999. – 127, № 2. – P. 409 – 415.
7. Leviatan D., Shevchuk I. A. Coconvex polynomial approximation // J. Approxim. Theory. – 2003. – 121,
№ 1. – P. 100 – 118.
8. Kopotun K. A. Pointwise and uniform estimates for convex approximation of functions by algebraic
polynomials // Constr. Approxim. – 1994. – 10, № 2. – P. 153 – 178.
9. Pleshakov M. G., Shatalina A. V. Piecewise coapproximation and the Whitney inequality // J. Approx.
Theory. – 2000. – 105, № 2. – P. 189 – 210.
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Received 26.05.09
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3
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| id | umjimathkievua-article-2873 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:31:59Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d7/c208ef65acd466000d2859c697dc59d7.pdf |
| spelling | umjimathkievua-article-28732020-03-18T19:39:19Z Are the degrees of the best (co)convex and unconstrained polynomial approximations the same? II Чи однакові порядки найкращого (ко)опуклого наближення та поліноміального наближення без обмежень? ІІ Kopotun, K. A. Leviatan, D. Shevchuk, I. A. Коротун, К. А. Левіатан, Д. Шевчук, І. О. In Part I of the paper, we have proved that, for every $α > 0$ and a continuous function $f$, which is either convex $(s = 0)$ or changes convexity at a finite collection $Y_s = \{y_i\}^s_i = 1$ of points $y_i ∈ (-1, 1)$, $$\sup \left\{n^{\alpha}E^{(2)}_n(f,Y_s):\;n \geq N^{*}\right\} \leq c(\alpha,s) \sup \left\{n^{\alpha}E_n(f):\; n \geq 1 \right\},$$ where $E_n (f)$ and $E^{(2)}_n (f, Y_s)$ denote, respectively, the degrees of the best unconstrained and (co)convex approximations and $c(α, s)$ is a constant depending only on $α$ and $s$. Moreover, it has been shown that $N^{∗}$ may be chosen to be 1 for $s = 0$ or $s = 1, α ≠ 4$, and that it must depend on $Y_s$ and $α$ for $s = 1, α = 4$ or $s ≥ 2$. In Part II of the paper, we show that a more general inequality $$\sup \left\{n^{\alpha}E^{(2)}_n(f,Y_s):\;n \geq N^{*}\right\} \leq c(\alpha, N, s) \sup \left\{n^{\alpha}E_n(f):\; n \geq N \right\},$$ is valid, where, depending on the triple $(α,N,s)$ the number $N^{∗}$ may depend on $α,N,Y_s$, and $f$ or be independent of these parameters. У частині І цієї статті доведено, що для кожного $α > 0$ та неперервної функції $f$, яка або опукла $(s = 0)$ або змінює опуклість у скінченному наборі $Y_s = \{y_i\}^s_i = 1$ точок $y_i ∈ (-1, 1)$, $$\sup \left\{n^{\alpha}E^{(2)}_n(f,Y_s):\;n \geq N^{*}\right\} \leq c(\alpha,s) \sup \left\{n^{\alpha}E_n(f):\; n \geq 1 \right\},$$ де $E_n (f)$ та $E^{(2)}_n (f, Y_s)$ означають відповідно порядок найкращого наближення без обмежень та (ко)опуклого наближення, $c(α, s)$ є сталою, що залежить лише від $α$ і $s$: Більш того, було показано, що $N^{∗}$ можна вибрати рівним одиниці, якщо $s = 0$ або $s = 1, α ≠ 4$, і що воно повинно залежати від $Y_s$ і $α$, якщо $s = 1, α = 4$4 або $s ≥ 2$. У частині II показано, що виконується більш загальна нерівність $$\sup \left\{n^{\alpha}E^{(2)}_n(f,Y_s):\;n \geq N^{*}\right\} \leq c(\alpha, N, s) \sup \left\{n^{\alpha}E_n(f):\; n \geq N \right\},$$ де в залежності від трійки $(α,N,s)$ число $N^{∗}$ може залежати або ні від $α,N,Y_s$ та $f$. Institute of Mathematics, NAS of Ukraine 2010-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2873 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 3 (2010); 369–386 Український математичний журнал; Том 62 № 3 (2010); 369–386 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2873/2497 https://umj.imath.kiev.ua/index.php/umj/article/view/2873/2498 Copyright (c) 2010 Kopotun K. A.; Leviatan D.; Shevchuk I. A. |
| spellingShingle | Kopotun, K. A. Leviatan, D. Shevchuk, I. A. Коротун, К. А. Левіатан, Д. Шевчук, І. О. Are the degrees of the best (co)convex and unconstrained polynomial approximations the same? II |
| title | Are the degrees of the best (co)convex and unconstrained polynomial approximations the same? II |
| title_alt | Чи однакові порядки найкращого (ко)опуклого наближення та поліноміального наближення без обмежень? ІІ |
| title_full | Are the degrees of the best (co)convex and unconstrained polynomial approximations the same? II |
| title_fullStr | Are the degrees of the best (co)convex and unconstrained polynomial approximations the same? II |
| title_full_unstemmed | Are the degrees of the best (co)convex and unconstrained polynomial approximations the same? II |
| title_short | Are the degrees of the best (co)convex and unconstrained polynomial approximations the same? II |
| title_sort | are the degrees of the best (co)convex and unconstrained polynomial approximations the same? ii |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2873 |
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