On moduli of smoothness with Jacobi weights
We introduce the moduli of smoothness with Jacobi weights $(1 x)\alpha (1+x)\beta$ for functions in the Jacobi weighted spaces $L_p[ 1, 1],\; 0 < p \leq \infty $. These moduli are used to characterize the smoothness of (the derivatives of) functions in the weighted spaces $L_p$. If $1 \leq...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507365677203456 |
|---|---|
| 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 | 2019-12-05T09:18:29Z |
| description | We introduce the moduli of smoothness with Jacobi weights $(1 x)\alpha (1+x)\beta$ for functions in the Jacobi weighted spaces
$L_p[ 1, 1],\; 0 < p \leq \infty $. These moduli are used to characterize the smoothness of (the derivatives of) functions in the
weighted spaces $L_p$. If $1 \leq p \leq \infty$, then these moduli are equivalent to certain weighted $K$-functionals (and so they are
equivalent to certain weighted Ditzian – Totik moduli of smoothness for these $p$), while for $0 < p < 1$ they are equivalent
to certain “Realization functionals”. |
| first_indexed | 2026-03-24T02:08:10Z |
| format | Article |
| fulltext |
UDC 517.5
K. A. Kopotun (Univ. Manitoba, Winnipeg, Canada),
D. Leviatan (Raymond and Beverly Sacker School Math. Sci., Tel Aviv Univ., Israel),
I. A. Shevchuk (Taras Shevchenko Nat. Univ., Kyiv, Ukraine)
ON MODULI OF SMOOTHNESS WITH JACOBI WEIGHTS*
ПРО МОДУЛI ГЛАДКОСТI З ВАГАМИ ЯКОБI
We introduce the moduli of smoothness with Jacobi weights (1 - x)\alpha (1+ x)\beta for functions in the Jacobi weighted spaces
Lp[ - 1, 1], 0 < p \leq \infty . These moduli are used to characterize the smoothness of (the derivatives of) functions in the
weighted spaces Lp . If 1 \leq p \leq \infty , then these moduli are equivalent to certain weighted K -functionals (and so they are
equivalent to certain weighted Ditzian – Totik moduli of smoothness for these p), while for 0 < p < 1 they are equivalent
to certain “Realization functionals”.
Введено модулi гладкостi з вагами Якобi (1 - x)\alpha (1 + x)\beta для функцiй, що належать ваговим просторам Якобi
Lp[ - 1, 1], 0 < p \leq \infty . Цi модулi використовуються, щоб охарактеризувати гладкiсть функцiй та їх похiдних у
вагових просторах Lp. При 1 \leq p \leq \infty цi модулi еквiвалентнi деяким ваговим K -функцiоналам (таким чином,
еквiвалентнi деяким ваговим модулям гладкостi Дiцiана – Тотiка для цих p). Водночас при 0 < p < 1 цi модулi
еквiвалентнi деяким „функцiоналам реалiзацiй”.
1. Introduction and main results. The main purpose of this paper is to introduce moduli of
smoothness with Jacobi weights (1 - x)\alpha (1 + x)\beta for functions in the Jacobi weighted Lp[ - 1, 1],
0 < p \leq \infty , spaces. These moduli generalize the moduli that were recently introduced by the authors
in [9, 10] in order to characterize the smoothness of (the derivatives of) functions in the ordinary
(unweighted) Lp spaces.
For a measurable function f : [ - 1, 1] \mapsto \rightarrow \BbbR and an interval I \subseteq [ - 1, 1], we use the usual
notation \| f\| Lp(I)
:=
\biggl( \int
I
| f(x)| p dx
\biggr) 1/p
, 0 < p < \infty , and \| f\| L\infty (I) := \mathrm{e}\mathrm{s}\mathrm{s} \mathrm{s}\mathrm{u}\mathrm{p}x\in I | f(x)| . For a
weight function w, we let Lw,p(I) := \{ f | \| wf\| Lp(I)
< \infty \} , and, for f \in Lw,p(I), we denote
by En(f, I)w,p := \mathrm{i}\mathrm{n}\mathrm{f}pn\in \BbbP n \| w(f - pn)\| Lp(I), the error of best weighted approximation of f by
polynomials in \BbbP n, the set of algebraic polynomials of degree strictly less than n. For I = [ - 1, 1],
we denote \| \cdot \| p := \| \cdot \| Lp[ - 1,1] , Lw,p := Lw,p[ - 1, 1], En(f)w,p := En(f, [ - 1, 1])w,p, etc. Finally,
denote
\varphi (x) :=
\sqrt{}
1 - x2.
Definition 1.1. For r \in \BbbN 0 and 0 < p \leq \infty , denote \BbbB 0
p(w) := Lw,p and
\BbbB r
p(w) :=
\Bigl\{
f | f (r - 1) \in ACloc( - 1, 1) and \varphi rf (r) \in Lw,p
\Bigr\}
, r \geq 1,
where ACloc( - 1, 1) denotes the set of functions which are locally absolutely continuous in ( - 1, 1).
Now, define
Jp :=
\left\{ ( - 1/p,\infty ), if p <\infty ,
[0,\infty ), if p =\infty ,
* Supported by NSERC of Canada.
c\bigcirc K. A. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK, 2018
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3 379
380 K. A. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
let
w\alpha ,\beta (x) := (1 - x)\alpha (1 + x)\beta , \alpha , \beta \in Jp,
be the Jacobi weights, and denote L\alpha ,\beta
p := Lw\alpha ,\beta ,p.
Also denote
\scrW \xi ,\zeta
\delta (x) := (1 - x - \delta \varphi (x)/2)\xi (1 + x - \delta \varphi (x)/2)\zeta .
Note that \scrW \alpha ,\beta
0 (x) = w\alpha ,\beta (x), \scrW
1/2,1/2
0 (x) = \varphi (x) and, if \xi , \zeta \geq 0, \scrW \xi ,\zeta
\delta (x) \leq w\xi ,\zeta (x).
For k \in \BbbN and h \geq 0, let
\Delta k
h(f, x; J) :=
\left\{
\sum k
i=0
\biggl(
k
i
\biggr)
( - 1)k - if
\biggl(
x - kh
2
+ ih
\biggr)
, if
\biggl[
x - kh
2
, x+
kh
2
\biggr]
\subseteq J,
0, otherwise,
be the kth symmetric difference, and \Delta k
h(f, x) := \Delta k
h(f, x; [ - 1, 1]).
We introduce the following definition, which for \alpha , \beta = 0, was given in [10] (Definition 2.2) (for
\alpha , \beta = 0 and p =\infty see the earlier [2] (Chapter 3.10)).
Definition 1.2. For k, r \in \BbbN and f \in \BbbB r
p(w\alpha ,\beta ), 0 < p \leq \infty , define
\omega \varphi
k,r(f
(r), t)\alpha ,\beta ,p := \mathrm{s}\mathrm{u}\mathrm{p}
0\leq h\leq t
\bigm\| \bigm\| \bigm\| \scrW r/2+\alpha ,r/2+\beta
kh (\cdot )\Delta k
h\varphi (\cdot )(f
(r), \cdot )
\bigm\| \bigm\| \bigm\|
p
. (1.1)
For \delta > 0, denote (see [10])
\frakD \delta :=
\bigl\{
x
\bigm| \bigm| 1 - \delta \varphi (x)/2 \geq | x|
\bigr\}
\setminus \{ \pm 1\} =
\biggl\{
x
\bigm| \bigm| | x| \leq 4 - \delta 2
4 + \delta 2
\biggr\}
= [ - 1 + \mu (\delta ), 1 - \mu (\delta )],
where
\mu (\delta ) := 2\delta 2/(4 + \delta 2).
Observe that \frakD \delta 1 \subset \frakD \delta 2 if \delta 2 < \delta 1 \leq 2, and that \frakD \delta = \varnothing if \delta > 2. Also note that \Delta k
h\varphi (x)(f, x) is
defined to be identically 0 if x \not \in \frakD kh and that \scrW r/2+\alpha ,r/2+\beta
\delta is well defined on \frakD \delta (except perhaps
at the endpoints where it may be infinite).
Hence,
\omega \varphi
k,r(f
(r), t)\alpha ,\beta ,p = \mathrm{s}\mathrm{u}\mathrm{p}
0<h\leq t
\bigm\| \bigm\| \bigm\| \scrW r/2+\alpha ,r/2+\beta
kh (\cdot )\Delta k
h\varphi (\cdot )(f
(r), \cdot )
\bigm\| \bigm\| \bigm\|
Lp(\frakD kh)
(1.2)
and
\omega \varphi
k,r(f
(r), t)\alpha ,\beta ,p = \omega \varphi
k,r(f
(r), 2/k)\alpha ,\beta ,p for t \geq 2/k. (1.3)
In a forthcoming paper [11], we will prove Whitney-, Jackson- and Bernstein-type theorems for
the Jacobi weighted approximation of functions in the above spaces by algebraic polynomials. Thus,
we get a constructive characterization of the smoothness classes with respect to these moduli by
means of the degrees of approximation. This implies, in particular, that these moduli are the right
measure of smoothness to be used while investigating constrained weighted approximation (see, e. g.,
[3, 7, 8]).
We will show that, for r/2+\alpha , r/2+\beta \geq 0, our moduli are equivalent to the following weighted
averaged moduli.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
ON MODULI OF SMOOTHNESS WITH JACOBI WEIGHTS 381
Definition 1.3. For k \in \BbbN , r \in \BbbN 0 and f \in \BbbB r
p(w\alpha ,\beta ), 0 < p <\infty , the kth weighted averaged
modulus of smoothness of f is defined as
\omega \ast \varphi
k,r(f
(r), t)\alpha ,\beta ,p :=
\left( 1
t
t\int
0
\int
\frakD k\tau
| \scrW r/2+\alpha ,r/2+\beta
k\tau (x)\Delta k
\tau \varphi (x)(f
(r), x)| p dx d\tau
\right)
1/p
.
If p =\infty and f \in \BbbB r
\infty (w\alpha ,\beta ), we write
\omega \ast \varphi
k,r(f
(r), t)\alpha ,\beta ,\infty := \omega \varphi
k,r(f
(r), t)\alpha ,\beta ,\infty .
Clearly,
\omega \ast \varphi
k,r(f
(r), t)\alpha ,\beta ,p \leq \omega \varphi
k,r(f
(r), t)\alpha ,\beta ,p, t > 0, 0 < p \leq \infty . (1.4)
We now define the weighted K -functional as well as the “Realization functional” as follows.
Definition 1.4. For k \in \BbbN , r \in \BbbN 0 and f \in \BbbB r
p(w\alpha ,\beta ), 0 < p \leq \infty , define
K\varphi
k,r(f
(r), tk)\alpha ,\beta ,p := \mathrm{i}\mathrm{n}\mathrm{f}
g\in \BbbB k+r
p (w\alpha ,\beta )
\biggl\{ \bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
r(f (r) - g(r))
\bigm\| \bigm\| \bigm\|
p
+ tk
\bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
k+rg(k+r)
\bigm\| \bigm\| \bigm\|
p
\biggr\}
and
R\varphi
k,r(f
(r), n - k)\alpha ,\beta ,p := \mathrm{i}\mathrm{n}\mathrm{f}
Pn\in \BbbP n
\biggl\{ \bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
r(f (r) - P (r)
n )
\bigm\| \bigm\| \bigm\|
p
+ n - k
\bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
k+rP (k+r)
n
\bigm\| \bigm\| \bigm\|
p
\biggr\}
.
Clearly, K\varphi
k,r(f
(r), n - k)\alpha ,\beta ,p \leq R\varphi
k,r(f
(r), n - k)\alpha ,\beta ,p, n \in \BbbN . Note that, as is rather well known,
K -functionals are not the right measure of smoothness if 0 < p < 1, since they may become
identically zero.
Throughout this paper, all constants c may depend only on k, r, p, \alpha and \beta , but are independent
of the function as well as the important parameters t and n. The constants c may be different even
if they appear in the same line.
Our first main result in this paper is the following theorem. It is a corollary of Lemma 3.2 and
the sequence of estimates (4.3).
Theorem 1.1. If k \in \BbbN , r \in \BbbN 0, r/2 + \alpha \geq 0, r/2 + \beta \geq 0, 1 \leq p \leq \infty , and f \in \BbbB r
p(w\alpha ,\beta ),
then there exists N \in \BbbN depending on k, r, p, \alpha and \beta , such that for all 0 < t \leq 2/k and n \in \BbbN
satisfying \mathrm{m}\mathrm{a}\mathrm{x}\{ N, c1/t\} \leq n \leq c2/t,
K\varphi
k,r(f
(r), tk)\alpha ,\beta ,p \leq cR\varphi
k,r(f
(r), n - k)\alpha ,\beta ,p \leq c\omega \ast \varphi
k,r(f
(r), t)\alpha ,\beta ,p \leq
\leq c\omega \varphi
k,r(f
(r), t)\alpha ,\beta ,p \leq cK\varphi
k,r(f
(r), tk)\alpha ,\beta ,p, (1.5)
where constants c may depend only on k, r, p, \alpha , \beta as well as c1 and c2.
Remark 1.1. Clearly, K\varphi
k,r(f
(r), tk)\alpha ,\beta ,p \leq
\bigm\| \bigm\| w\alpha ,\beta \varphi
rf (r)
\bigm\| \bigm\|
p
< \infty , for all f \in \BbbB r
p(w\alpha ,\beta ), and it
follows from Theorem 2.1 that, if r/2 + \alpha < 0 or/and r/2 + \beta < 0, then there exists a function
f \in \BbbB r
p(w\alpha ,\beta ) such that \omega \varphi
k,r(f
(r), t)\alpha ,\beta ,p = \infty , for all t > 0. Hence, Theorem 1.1 is not valid if
r/2 + \alpha < 0 or/and r/2 + \beta < 0.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
382 K. A. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
We can somewhat simplify the statement of Theorem 1.1 if we remove the realization functional
R\varphi
k,r from (1.5).
Corollary 1.1. If k \in \BbbN , r \in \BbbN 0, r/2 + \alpha \geq 0, r/2 + \beta \geq 0, 1 \leq p \leq \infty , and f \in \BbbB r
p(w\alpha ,\beta ),
then, for all 0 < t \leq 2/k,
K\varphi
k,r(f
(r), tk)\alpha ,\beta ,p \leq c\omega \ast \varphi
k,r(f
(r), t)\alpha ,\beta ,p \leq c\omega \varphi
k,r(f
(r), t)\alpha ,\beta ,p \leq cK\varphi
k,r(f
(r), tk)\alpha ,\beta ,p.
In the case 0 < p < 1, we have the following result on the equivalence of the moduli and
Realization functionals. It is a corollary of Theorem 4.3 that will be proved in Section 4.
Theorem 1.2. Let k \in \BbbN , r \in \BbbN 0, 0 < p < 1, r/2 + \alpha \geq 0, r/2 + \beta \geq 0, and f \in \BbbB r
p(w\alpha ,\beta ).
Then there exist N \in \BbbN and \vargamma > 0 depending on k, p, \alpha and \beta , such that, for any \vargamma 1 \in (0, \vargamma ],
n \geq N, \vargamma 1/n \leq t \leq \vargamma /n, we have
R\varphi
k,r(f, n
- k)\alpha ,\beta ,p \sim \omega \ast \varphi
k,r(f
(r), t)\alpha ,\beta ,p \sim \omega \varphi
k,r(f
(r), t)\alpha ,\beta ,p.
Here, as usual, by a(t) \sim b(t), t \in T, we mean that there exists a positive constant c0 such that
c - 1
0 a(t) \leq b(t) \leq c0a(t), for all t \in T.
Note that it follows from Theorem 1.2 that, for sufficiently small t1, t2 > 0 such that t1 \sim t2,
\omega \ast \varphi
k,r(f
(r), t1)\alpha ,\beta ,p \sim \omega \varphi
k,r(f
(r), t1)\alpha ,\beta ,p \sim \omega \ast \varphi
k,r(f
(r), t2)\alpha ,\beta ,p \sim \omega \varphi
k,r(f
(r), t2)\alpha ,\beta ,p.
If 1 \leq p \leq \infty , we can say a bit more. Theorem 1.1 and the (obvious) monotonicity of \omega \varphi
k,r(f
(r), t)\alpha ,\beta ,p,
with respect to t, immediately yield the following quite useful property which is not easily seen from
Definition 1.2.
Corollary 1.2. Let k \in \BbbN , r \in \BbbN 0, r/2 + \alpha \geq 0, r/2 + \beta \geq 0, 1 \leq p \leq \infty , f \in \BbbB r
p(w\alpha ,\beta ) and
\lambda \geq 1. Then, for all t > 0,
\omega \varphi
k,r(f
(r), \lambda t)\alpha ,\beta ,p \leq c\lambda k\omega \varphi
k,r(f
(r), t)\alpha ,\beta ,p. (1.6)
By virtue of (5.2) the following result is an immediate consequence of Corollary 1.1.
Theorem 1.3. Let k \in \BbbN , r \in \BbbN 0, r/2+\alpha \geq 0, r/2+\beta \geq 0, and 1 \leq p \leq \infty . If f \in \BbbB r
p(w\alpha ,\beta ),
then, for some t0 > 0 independent of f,
\omega \varphi
k,r(f
(r), t)\alpha ,\beta ,p \sim \omega k
\varphi (f
(r), t)w\alpha ,\beta \varphi r,p, 0 < t \leq t0, (1.7)
where the weighted DT moduli \omega k
\varphi (g, \cdot )w,p are defined in (5.1).
It was shown in [9] (Theorem 5.1) that, for \xi , \zeta \geq 0 and g \in B1
p(w\xi ,\zeta ),
\omega k+1
\varphi (g, t)w\xi ,\zeta ,p \leq ct\omega k
\varphi (g
\prime , t)w\xi ,\zeta \varphi ,p, t > 0.
Letting \xi := r/2 + \alpha , \zeta := r/2 + \beta , g := f (r), using the fact that f (r) \in B1
p(wr/2+\alpha ,r/2+\beta ) if
and only if f \in Br+1
p (w\alpha ,\beta ), by virtue of (1.7), as well as (1.6) if t is “large” (i.e., if t > t0), we
immediately get the following result.
Lemma 1.1. Let k \in \BbbN , r \in \BbbN 0, r/2+\alpha \geq 0, r/2+\beta \geq 0, and 1 \leq p \leq \infty . If f \in \BbbB r+1
p (w\alpha ,\beta ),
then
\omega \varphi
k+1,r(f
(r), t)\alpha ,\beta ,p \leq ct\omega \varphi
k,r+1(f
(r+1), t)\alpha ,\beta ,p, t > 0.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
ON MODULI OF SMOOTHNESS WITH JACOBI WEIGHTS 383
Finally, the following lemma follows from [1] (Theorem 6.1.4) using (1.7).
Lemma 1.2. Let k \in \BbbN , r \in \BbbN 0, r/2+\alpha \geq 0, r/2+\beta \geq 0, and 1 \leq p \leq \infty . If f \in \BbbB r
p(w\alpha ,\beta ),
then
\omega \varphi
k+1,r(f
(r), t)\alpha ,\beta ,p \leq c\omega \varphi
k,r(f
(r), t)\alpha ,\beta ,p, t > 0.
2. Hierarchy of \bfitB \bfitr
\bfitp (\bfitw \bfitalpha ,\bfitbeta ), (un)boundedness of the moduli and their convergence to \bfzero .
Without special references we use the following evident inequalities:
(1 - x) \leq 2(1 - u) and (1 + x) \leq 2(1 + u), if u \in [\mathrm{m}\mathrm{i}\mathrm{n}\{ 0, x\} ,\mathrm{m}\mathrm{a}\mathrm{x}\{ 0, x\} ] ,
and
\varphi (x) \leq \varphi (u), if | u| \leq | x| \leq 1.
Also (see [10], Proposition 3.1(iv)),
| \varphi \prime (x)| \leq 1/\delta for x \in \frakD \delta . (2.1)
First we show the hierarchy between the \BbbB r
p(w\alpha ,\beta ), r \geq 0, spaces. Namely, the following lemma
holds.
Lemma 2.1. Let r \in \BbbN 0, 1 \leq p \leq \infty and r/2 + \alpha , r/2 + \beta \in Jp. Then
\BbbB r+1
p (w\alpha ,\beta ) \subseteq \BbbB r
p(w\alpha ,\beta ). (2.2)
Moreover, in the case p =\infty , if r/2 + \alpha > 0 and r/2 + \beta > 0, then, additionally,
f \in \BbbB r+1
\infty (w\alpha ,\beta ) =\Rightarrow \mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow \pm 1
w\alpha ,\beta (x)\varphi
r(x)f (r)(x) = 0. (2.3)
Remark 2.1. Note that we may not relax the condition r/2+\alpha , r/2+\beta > 0 in order to guarantee
(2.3). Indeed, if \alpha = - r/2, for example, then the function g(x) := xr is certainly in \BbbB r+1
\infty (w\alpha ,\beta )
but \mathrm{l}\mathrm{i}\mathrm{m}x\rightarrow 1w\alpha ,\beta (x)\varphi
r(x)g(r)(x) \not = 0.
The same example shows that we may not relax the condition r/2 + \alpha , r/2 + \beta \in Jp in order to
guarantee (2.2), since
\bigm\| \bigm\| w\alpha ,\beta \varphi
rg(r)
\bigm\| \bigm\|
p
=\infty if this condition is not satisfied, so that g \not \in \BbbB r
p(w\alpha ,\beta ).
Remark 2.2. For any r \in \BbbN 0 and \alpha , \beta \in \BbbR , (2.2) is not valid if 0 < p < 1. For example,
suppose that f is such that
f (r)(x) =
\infty \sum
n=1
gn(x),
where, for each n \in \BbbN ,
gn(x) :=
\left\{
Hn
\varepsilon n
\biggl(
x+ 1 - 1
n+ 1
\biggr)
, if
1
n+ 1
< x+ 1 \leq 1
n+ 1
+ \varepsilon n,
Hn, if
1
n+ 1
+ \varepsilon n < x+ 1 \leq 1
n
- \varepsilon n,
Hn
\varepsilon n
\biggl(
1
n
- x - 1
\biggr)
, if
1
n
- \varepsilon n < x+ 1 \leq 1
n
,
0, otherwise,
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
384 K. A. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
Hn := nr/2+\beta +1/p, \varepsilon n := c0n
- 2/(1 - p), and c0 > 0 is a constant depending only on p that guarantees
that 4\varepsilon nn(n+ 1) < 1, for all n \in \BbbN . Then f (r) \in \mathrm{A}\mathrm{C}loc( - 1, 1) and
\bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
rf (r)
\bigm\| \bigm\| \bigm\| p
p
=
\infty \sum
n=1
\| w\alpha ,\beta \varphi
rgn\| pp \geq c
\infty \sum
n=1
1
n(r/2+\beta )p
Hp
nn
- 2 = c
\infty \sum
n=1
n - 1 =\infty .
Hence, f \not \in Br
p(w\alpha ,\beta ). At the same time,
\bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
r+1f (r+1)
\bigm\| \bigm\| \bigm\| p
p
=
\infty \sum
n=1
\bigm\| \bigm\| w\alpha ,\beta \varphi
r+1g\prime n
\bigm\| \bigm\| p
p
\leq c
\infty \sum
n=1
1
n((r+1)/2+\beta )p
\bigl(
Hn\varepsilon
- 1
n
\bigr) p
\varepsilon n =
= c
\infty \sum
n=1
n1 - p/2\varepsilon 1 - p
n = c
\infty \sum
n=1
n - 1 - p/2 <\infty ,
so that f \in Br+1
p (w\alpha ,\beta ).
Proof of Lemma 2.1. The proof follows along the lines of [10] (Lemma 3.4) with some
modifications, we bring it here for the sake of completeness. Let g \in \BbbB r+1
p (w\alpha ,\beta ), and assume,
without loss of generality, that g(r)(0) = 0 and that \beta \geq \alpha . For convenience, denote Ap :=
:=
\bigm\| \bigm\| w\alpha ,\beta \varphi
r+1g(r+1)
\bigm\| \bigm\|
p
.
First, if p =\infty , then A\infty <\infty and
w\alpha ,\beta (x)\varphi
r(x)
\bigm| \bigm| g(r)(x)\bigm| \bigm| = w\alpha ,\beta (x)\varphi
r(x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
x\int
0
g(r+1)(u) du
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq A\infty w\alpha ,\beta (x)\varphi
r(x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
x\int
0
w - 1
\alpha ,\beta (u)\varphi
- r - 1(u) du
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq 2\beta - \alpha A\infty \varphi r+2\alpha (x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
x\int
0
\varphi - r - 1 - 2\alpha (u) du
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
= 2\beta - \alpha A\infty \varphi r+2\alpha (x)
| x| \int
0
\varphi - r - 1 - 2\alpha (u) du \leq
\leq 2\beta - \alpha A\infty
| x| \int
0
\varphi - 1(u) du \leq
\leq 2\beta - \alpha A\infty
1\int
0
\varphi - 1(u) du = \pi 2\beta - \alpha - 1A\infty .
Hence, g \in \BbbB r
\infty (w\alpha ,\beta ), and (2.2) is proved if p =\infty .
In order to prove (2.3) we need to show that, if r/2 + \alpha , r/2 + \beta > 0, then
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ON MODULI OF SMOOTHNESS WITH JACOBI WEIGHTS 385
\mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow \pm 1
w\alpha ,\beta (x)\varphi
r(x)g(r)(x) = 0. (2.4)
(Note that we are still not losing generality by assuming that g(r)(0) = 0.) We put \varepsilon := \mathrm{m}\mathrm{i}\mathrm{n}\{ r +
+ 2\alpha , 1\} > 0 and note that
x\int
0
1
\varphi 2(u)
du =
1
2
\mathrm{l}\mathrm{n}
1 + x
1 - x
.
Therefore,
w\alpha ,\beta (x)\varphi
r(x)| g(r)(x)| \leq 2\beta - \alpha A\infty \varphi \varepsilon (x)
| x| \int
0
1
\varphi 1+\varepsilon (u)
du \leq
\leq 2\beta - \alpha A\infty \varphi \varepsilon (x)
| x| \int
0
1
\varphi 2(u)
du =
= 2\beta - \alpha A\infty \varphi \varepsilon (| x| ) \mathrm{l}\mathrm{n} 1 + | x|
1 - | x|
\rightarrow 0, | x| \rightarrow 1,
and (2.4) is proved.
Now let 1 \leq p <\infty and q := p/(p - 1). Then, denoting\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
x\int
0
| G(u)| qdu
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1/q
:= \mathrm{s}\mathrm{u}\mathrm{p}
u\in [min\{ 0,x\} ,max\{ 0,x\} ]
| G(u)|
if q =\infty , we have by Hölder’s inequality
\bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
rg(r)
\bigm\| \bigm\| \bigm\| p
p
=
1\int
- 1
wp
\alpha ,\beta (x)\varphi
rp(x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
x\int
0
g(r+1)(u) du
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
p
dx \leq
\leq
1\int
- 1
wp
\alpha ,\beta (x)\varphi
rp(x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
x\int
0
w - q
\alpha ,\beta (u)\varphi
- (r+1)q(u) du
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
p/q
\times
\times
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
x\int
0
| w\alpha ,\beta (u)\varphi
r+1(u)g(r+1)(u)| p du
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| dx \leq
\leq Ap
p
1\int
- 1
wp
\alpha ,\beta (x)\varphi
rp(x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
x\int
0
w - q
\alpha ,\beta (u)\varphi
- (r+1)q(u)du
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
p/q
dx \leq
\leq 2(\beta - \alpha )pAp
p
1\int
- 1
\varphi rp+2\alpha p(x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
x\int
0
\varphi - (r+1)q - 2\alpha q(u)du
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
p/q
dx =:
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386 K. A. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
=: 2(\beta - \alpha )pAp
p\Theta (\alpha , p).
Note that
\Theta (\alpha , 1) =
1\int
- 1
\varphi r+2\alpha (x)
\Biggl(
\mathrm{s}\mathrm{u}\mathrm{p}
u\in [min\{ 0,x\} ,max\{ 0,x\} ]
\varphi - r - 1 - 2\alpha (u)
\Biggr)
dx.
Recall that r/2 + \alpha \in Jp so that rp+ 2\alpha p > - 2. We consider two cases.
Case 1. Suppose that rp + 2\alpha p \geq - 1. If p = 1, then r + 2\alpha + 1 \geq 0 implies that \Theta (\alpha , 1) =
=
\int 1
- 1
\varphi - 1(x)dx = \pi , and if 1 < p <\infty , then ((r + 1)q - 1 + 2\alpha q)p/q = rp+ 2\alpha p+ 1 \geq 0, and
hence
\Theta (\alpha , p) = 2
1\int
0
1
\varphi (x)
\left( x\int
0
\varphi (r+1)q - 1+2\alpha q(x)
\varphi (r+1)q+2\alpha q(u)
du
\right) p/q dx \leq
\leq 2
1\int
0
1
\varphi (x)
\left( x\int
0
1
\varphi (u)
du
\right) p/q dx \leq 2
1\int
0
dx
\varphi (x)
\left( 1\int
0
du
\varphi (u)
\right) p/q =
= 2(\pi /2)p.
Case 2. Suppose now that - 2 < rp+ 2\alpha p < - 1. If p = 1, then
\Theta (\alpha , 1) =
1\int
- 1
\varphi r+2\alpha (x)dx <\infty .
If 1 < p <\infty , then (r + 1)q + 2\alpha q < 1. Hence
1\int
0
\varphi - (r+1)q - 2\alpha q(u)du <
1\int
0
\varphi - 1(u)du = \pi /2,
and so
\Theta (\alpha , p) \leq 2(\pi /2)p/q
1\int
0
\varphi rp+2\alpha p(x)dx <\infty .
Lemma 2.1 is proved.
We now show that, for a function f \in \BbbB r
p(w\alpha ,\beta ), if r/2 + \alpha \geq 0 and r/2 + \beta \geq 0, then the
modulus \omega \varphi
k,r(f
(r), t)\alpha ,\beta , p is bounded.
Lemma 2.2. Let k \in \BbbN , r \in \BbbN 0, r/2+\alpha \geq 0, r/2+\beta \geq 0, and 0 < p \leq \infty . If f \in \BbbB r
p(w\alpha ,\beta ),
then
\omega \varphi
k,r(f
(r), t)\alpha ,\beta , p \leq c
\bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
rf (r)
\bigm\| \bigm\| \bigm\|
p
, t > 0, (2.5)
where c depends only on k and p.
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ON MODULI OF SMOOTHNESS WITH JACOBI WEIGHTS 387
Proof. In view of (1.3), we may limit ourselves to t \leq 2/k, and so \frakD kh \not = \varnothing if 0 < h \leq t. We
set
ui(x) := x+ (i - k/2)h\varphi (x), 0 \leq i \leq k,
and note that, for x \in \frakD kh,
Br(x) :=
\scrW r/2+\alpha ,r/2+\beta
kh (x)
w\alpha ,\beta (ui(x))\varphi r(ui(x))
=
=
\biggl(
1 - ui(x) - (k - i)h\varphi (x)
1 - ui(x)
\biggr) r/2+\alpha \biggl( 1 + ui(x) - ih\varphi (x)
1 + ui(x)
\biggr) r/2+\beta
\leq 1.
Therefore, \bigm\| \bigm\| \bigm\| \scrW r/2+\alpha ,r/2+\beta
kh (\cdot )f (r)(ui(\cdot ))
\bigm\| \bigm\| \bigm\|
L\infty (\frakD kh)
=
=
\bigm\| \bigm\| \bigm\| Br(\cdot )w\alpha ,\beta (ui(\cdot ))\varphi r(ui(\cdot ))f (r)(ui(\cdot ))
\bigm\| \bigm\| \bigm\|
L\infty (\frakD kh)
\leq
\leq
\bigm\| \bigm\| \bigm\| w\alpha ,\beta (ui(\cdot ))\varphi r(ui(\cdot ))f (r)(ui(\cdot ))
\bigm\| \bigm\| \bigm\|
L\infty (\frakD kh)
\leq
\bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
rf (r)
\bigm\| \bigm\| \bigm\|
\infty
,
that yields (2.5) for p =\infty .
To apply the same arguments to the case 0 < p <\infty we note that (2.1) yields | \varphi \prime (x)| \leq 1/(kh)
for x \in \frakD kh, so that
u\prime i(x) \geq 1 - | i - k/2| h| \varphi \prime (x)| \geq 1 - kh| \varphi \prime (x)| /2 \geq 1/2, x \in \frakD kh,
which implies \int
\frakD kh
| F (ui(x))| dx \leq 2
1\int
- 1
| F (u)| du
for each F \in L1[ - 1, 1].
Hence,\bigm\| \bigm\| \bigm\| \scrW r/2+\alpha ,r/2+\beta
kh (\cdot )f (r)(ui(\cdot ))
\bigm\| \bigm\| \bigm\| p
Lp(\frakD kh)
\leq
\bigm\| \bigm\| \bigm\| w\alpha ,\beta (ui(\cdot ))\varphi r(ui(\cdot ))f (r)(ui(\cdot ))
\bigm\| \bigm\| \bigm\| p
Lp(\frakD kh)
\leq
\leq 2
1\int
- 1
| w\alpha ,\beta (x)\varphi
r(x)f (r)(x)| pdx = 2
\bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
rf (r)
\bigm\| \bigm\| \bigm\| p
p
.
Thus,
\omega \varphi
k,r(f
(r), t)\alpha ,\beta , p \leq c \mathrm{m}\mathrm{a}\mathrm{x}
0\leq i\leq k
\bigm\| \bigm\| \bigm\| \scrW r/2+\alpha ,r/2+\beta
kh (\cdot )f (r)(ui(\cdot ))
\bigm\| \bigm\| \bigm\|
Lp(\frakD kh)
\leq c
\bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
rf (r)
\bigm\| \bigm\| \bigm\|
p
.
Lemma 2.2 is proved.
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388 K. A. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
Remark 2.3. The same proof yields a local version of (2.5) as well. Namely, for each h > 0
and [a, b] \subseteq \frakD kh,\bigm\| \bigm\| \bigm\| \scrW r/2+\alpha ,r/2+\beta
kh (\cdot )\Delta k
h\varphi (\cdot )(f
(r), \cdot )
\bigm\| \bigm\| \bigm\|
Lp[a,b]
\leq c
\bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
rf (r)
\bigm\| \bigm\| \bigm\|
Lp(S)
,
where S := [a - kh\varphi (a)/2, b+ kh\varphi (b)/2] .
We now show that the modulus \omega \varphi
k,r(f
(r), t)\alpha ,\beta ,p may be infinite for a function f \in \BbbB r
p(w\alpha ,\beta ) if
either r/2 + \alpha < 0 or r/2 + \beta < 0.
When p = \infty , this is obvious. Indeed, suppose that r/2 + \beta \geq 0 and - k \leq r/2 + \alpha <
< 0, and let f(x) := (x - 1)k+r. Then f \in \BbbB r
\infty (w\alpha ,\beta ) and \Delta k
h\varphi (x)(f
(r), x) \equiv chk\varphi k(x). Hence,
\scrW r/2+\alpha ,r/2+\beta
kh (x)\Delta k
h\varphi (x)(f
(r), x) \rightarrow \infty for x such that 1 - x - kh\varphi (x)/2 \rightarrow 0. This implies that
\omega \varphi
k,r(f
(r), t)\alpha ,\beta ,\infty = \infty for all t > 0. Note also that, by considering f \in Cr[ - 1, 1] such that
f(x) = (1 - | x| )k+r, x \not \in [ - 1/2, 1/2], one can easily see that the same conclusion holds if both
r/2 + \alpha and /2 + \beta are in [ - k, 0).
When p < \infty , the arguments are not so obvious, but the conclusion is the same. The following
theorem is valid.
Theorem 2.1. Suppose that k \in \BbbN , r \in \BbbN 0, \alpha \in \BbbR , 0 < p < \infty , and r/2 + \beta < 0. If
0 < p < 1 and r \geq 1, we additionally assume that r/2 + \beta < 1 - 1/p. Then there exists a function
f \in \BbbB r
p(w\alpha ,\beta ), such that, for all t > 0,
\omega \varphi
k,r(f
(r), t)\alpha ,\beta ,p =\infty .
Proof. Let \{ \varepsilon n\} \infty n=0 be a decreasing sequence of positive numbers, tending to zero, such that
\varepsilon 0 < 1/(2k) and
(2 + k)\varepsilon n < \varepsilon n - 1, n \in \BbbN .
Define
Jn :=
\bigl[
- 1 + \varepsilon n, - 1 + \varepsilon n(1 + 2 - n)
\bigr]
.
Now, let f be such that
f (r)(x) :=
\left\{ (x+ 1 - \varepsilon n)
- r/2 - \beta - 1/p , if x \in Jn for some n \in \BbbN ,
0, otherwise.
Note that, in the case r \geq 1, since - r/2 - \beta - 1/p+1 > 0, the function f (r - 1)(x) =
\int x
0
f (r)(u)du
is locally absolutely continuous on ( - 1, 1).
Now,
2 - | r/2+\alpha | p
\bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
rf (r)
\bigm\| \bigm\| \bigm\| p
p
\leq
\infty \sum
n=1
\int
Jn
| (1 + x)r/2+\beta f (r)(x)| pdx \leq
\leq
\infty \sum
n=1
\varepsilon (r/2+\beta )p
n
\int
Jn
| f (r)(x)| pdx =
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ON MODULI OF SMOOTHNESS WITH JACOBI WEIGHTS 389
=
\infty \sum
n=1
\varepsilon (r/2+\beta )p
n
\varepsilon n2 - n\int
0
t - (r/2+\beta )p - 1dt \leq
\leq c
\infty \sum
n=1
2(r/2+\beta )np <\infty .
Hence, f \in \BbbB r
p(w\alpha ,\beta ).
We now let
xn := - 1 + k
2
\varepsilon n, hn :=
\varepsilon n
\varphi (xn)
, and Ik,n := [xn, xn + \varepsilon n],
so that
\frakD khn = [xn, - xn] and hn <
\surd
2\varepsilon n \rightarrow 0, n\rightarrow \infty .
Since \varphi (x) \geq \varphi (xn), | x| \leq | xn| , we conclude that, for any x \in Ik,n \subset [xn, - xn],
x -
\biggl(
k
2
- 2
\biggr)
hn\varphi (x) = x - k
2
hn\varphi (x) + 2hn\varphi (x) \geq - 1 + 2hn\varphi (x) \geq
\geq - 1 + 2hn\varphi (xn) = - 1 + 2\varepsilon n > - 1 + \varepsilon n(1 + 2 - n).
Now, since \varphi is concave and \varphi ( - 1) = 0, we have
\varphi (xn + \varepsilon n) <
xn + \varepsilon n + 1
xn + 1
\varphi (xn) =
\biggl(
1 +
2
k
\biggr)
\varphi (xn),
and so, for all x \in Ik,n,
x+
k
2
hn\varphi (x) \leq xn + \varepsilon n +
k
2
hn\varphi (xn + \varepsilon n) \leq xn + \varepsilon n +
\biggl(
1 +
k
2
\biggr)
hn\varphi (xn) =
= - 1 + (2 + k)\varepsilon n < - 1 + \varepsilon n - 1.
If k \geq 2, this implies that, for all 2 \leq i \leq k and x \in Ik,n,
f (r)(x+ (i - k/2)hn\varphi (x)) = 0.
Now, denote
y(x) := x+ (1 - k/2)hn\varphi (x)
and observe that
1
2
< y\prime (x) <
3
2
, x \in [xn, - xn], (2.6)
since, if | x| \leq | xn| , then it follows from (2.1) that
hn| \varphi \prime (x)| < 1/k (2.7)
and so
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390 K. A. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
| y\prime (x) - 1| \leq k
2
hn| \varphi \prime (x)| < 1
2
.
For all k \in \BbbN , using \| f1 + f2\| p \leq \mathrm{m}\mathrm{a}\mathrm{x}\{ 1, 21/p - 1\}
\Bigl(
\| f1\| p + \| f2\| p
\Bigr)
, we obtain
2| \alpha +r/2|
\bigm\| \bigm\| \bigm\| \scrW r/2+\alpha ,r/2+\beta
khn
(\cdot )\Delta k
hn\varphi (f
(r), \cdot )
\bigm\| \bigm\| \bigm\|
p
\geq
\geq 2| \alpha +r/2|
\bigm\| \bigm\| \bigm\| \scrW r/2+\alpha ,r/2+\beta
khn
(\cdot )\Delta k
hn\varphi (f
(r), \cdot )
\bigm\| \bigm\| \bigm\|
Lp(Ik,n)
\geq
\geq
\bigm\| \bigm\| \bigm\| (1 + y(\cdot ) - hn\varphi (\cdot ))r/2+\beta
\Bigl(
f (r)(y(\cdot ) - hn\varphi (\cdot )) - kf (r)(y(\cdot ))
\Bigr) \bigm\| \bigm\| \bigm\|
Lp(Ik,n)
\geq
\geq k\mathrm{m}\mathrm{i}\mathrm{n}\{ 1, 21 - 1/p\}
\bigm\| \bigm\| \bigm\| (1 + y(\cdot ) - hn\varphi (\cdot ))r/2+\beta f (r)(y(\cdot ))
\bigm\| \bigm\| \bigm\|
Lp(Ik,n)
- c
\bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
rf (r)
\bigm\| \bigm\| \bigm\|
p
\geq
\geq k\mathrm{m}\mathrm{i}\mathrm{n}\{ 1, 21 - 1/p\}
\bigm\| \bigm\| \bigm\| (1 + y(\cdot ) - \varepsilon n)
r/2+\beta f (r)(y(\cdot ))
\bigm\| \bigm\| \bigm\|
Lp(Ik,n)
- c
\bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
rf (r)
\bigm\| \bigm\| \bigm\|
p
,
where, in the second last inequality, we used the fact that y\prime (x) - hn\varphi
\prime (x) = 1 - khn\varphi
\prime (x)/2 \sim 1
that follows from (2.7), and in the last inequality, we used that r/2 + \beta < 0 and that \varepsilon n \leq hn\varphi (x)
for all x \in [xn, - xn].
In order to complete the proof, we show that
H :=
\bigm\| \bigm\| \bigm\| (1 + y(\cdot ) - \varepsilon n)
r/2+\beta f (r)(y(\cdot ))
\bigm\| \bigm\| \bigm\|
Lp(Ik,n)
=\infty .
Assume to the contrary that H <\infty . Since
y(xn) = - 1 + \varepsilon n \leq y(x) < - 1 + \varepsilon n - 1, x \in Ik,n,
there is a positive number an \leq \varepsilon n, such that
f (r)(y(x)) = (1 - \varepsilon n + y(x)) - r/2 - \beta - 1/p, x \in [xn, xn + an].
Therefore,
Hp \geq
xn+an\int
xn
(1 - \varepsilon n + y(x)) - 1dx.
Using the change of variable v = u(x) := 1 - \varepsilon n + y(x) and (2.6) we get
Hp \geq 2
3
xn+an\int
xn
(u(x)) - 1u\prime (x)dx =
2
3
u(xn+an)\int
0
dv
v
=\infty ,
that contradicts our assumption H <\infty .
Thus, we have found a sequence \{ hn\} \infty n=0 of positive numbers, tending to zero, such that\bigm\| \bigm\| \bigm\| \scrW r/2+\alpha ,r/2+\beta
khn
\Delta k
hn\varphi
(f (r), \cdot )
\bigm\| \bigm\| \bigm\|
p
= \infty for all n \in \BbbN . This means that \omega \varphi
k,r(f
(r), t)\alpha ,\beta ,p = \infty for all
t > 0.
Theorem 2.1 is proved.
We now state some properties of the Jacobi weights that we need in several proofs below.
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ON MODULI OF SMOOTHNESS WITH JACOBI WEIGHTS 391
Proposition 2.1. For any \alpha , \beta \in \BbbR , x \in \frakD 2\delta and u \in [x - \delta \varphi (x)/2, x+ \delta \varphi (x)/2],
2 - | \alpha | - | \beta | w\alpha ,\beta (u) \leq w\alpha ,\beta (x) \leq 2| \alpha | +| \beta | w\alpha ,\beta (u), (2.8)
in particular,
\varphi (u)/2 \leq \varphi (x) \leq 2\varphi (u). (2.9)
Also,
2 - | \alpha | - | \beta | w\alpha ,\beta (x) \leq \scrW \alpha ,\beta
\delta (x) \leq 2| \alpha | +| \beta | w\alpha ,\beta (x), x \in \frakD 2\delta . (2.10)
Proof. For x \in \frakD 2\delta and u \in [x - \delta \varphi (x)/2, x+ \delta \varphi (x)/2], we have
(1 - u)/2 \leq (1 - x+ \delta \varphi (x)/2)/2 \leq 1 - x \leq 2(1 - x - \delta \varphi (x)/2) \leq 2(1 - u)
and
(1 + u)/2 \leq (1 + x+ \delta \varphi (x)/2)/2 \leq 1 + x \leq 2(1 + x - \delta \varphi (x)/2) \leq 2(1 + u).
This immediately yields (2.8). Now,
\scrW \alpha ,\beta
\delta (x) = w\alpha ,0(x+ \delta \varphi (x)/2)w0,\beta (x - \delta \varphi (x)/2) \leq
\leq 2| \alpha | w\alpha ,0(x)2
| \beta | w0,\beta (x) = 2| \alpha | +| \beta | w\alpha ,\beta (x)
and
w\alpha ,\beta (x) = w\alpha ,0(x)w0,\beta (x) \leq 2| \alpha | w\alpha ,0(x+ \delta \varphi (x)/2)2| \beta | w0,\beta (x - \delta \varphi (x)/2) =
= 2| \alpha | +| \beta | \scrW \alpha ,\beta
\delta (x).
Proposition 2.1 is proved.
Lemma 2.3. If k \in \BbbN , r \in \BbbN 0, r/2 + \alpha \geq 0, r/2 + \beta \geq 0, 0 < p < \infty , and f \in \BbbB r
p(w\alpha ,\beta ),
then
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow 0+
\omega \varphi
k,r(f
(r), t)\alpha ,\beta ,p = 0.
Proof. Let \epsilon > 0. For convenience, denote Cp := \mathrm{m}\mathrm{a}\mathrm{x}\{ 1, 21/p - 1\} . Since f \in \BbbB r
p(w\alpha ,\beta ), there
is \delta > 0 such that \bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
rf (r)
\bigm\| \bigm\| \bigm\|
Lp([ - 1,1]\setminus \frakD \delta )
<
\epsilon
2c0Cp
,
where c0 is the constant c from the statement of Lemma 2.2. Set
g(r)(x) :=
\left\{ f (r)(x), if x \in \frakD \delta ,
0, otherwise,
and note that, since g(r) \in Lp[ - 1, 1], there exists t0 > 0 such that
\omega \varphi
k (g
(r), t)p < \epsilon /(2| \alpha - \beta | +1Cp), 0 < t \leq t0.
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392 K. A. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
Using Lemma 2.2 and the fact that, if r/2+\alpha , r/2+ \beta \geq 0 and x \in \frakD kh, then \scrW r/2+\alpha ,r/2+\beta
kh (x) \leq
\leq 2| \alpha - \beta | , we have
\omega \varphi
k,r(f
(r), t)\alpha ,\beta ,p \leq Cp\omega
\varphi
k,r(g
(r), t)\alpha ,\beta ,p + Cp\omega
\varphi
k,r(f
(r) - g(r), t)\alpha ,\beta ,p \leq
\leq 2| \alpha - \beta | Cp\omega
\varphi
k (g
(r), t)p + c0Cp
\bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
r
\Bigl(
f (r) - g(r)
\Bigr) \bigm\| \bigm\| \bigm\|
p
<
< \epsilon /2 + c0Cp
\bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
rf (r)
\bigm\| \bigm\| \bigm\|
Lp([ - 1,1]\setminus \frakD \delta )
\leq \epsilon ,
if 0 < t \leq t0.
Lemma 2.3 is proved.
We now turn our attention to the case p =\infty . It is clear that, in order for
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow 0+
\omega \varphi
k,r(f
(r), t)\alpha ,\beta ,\infty = 0
to hold we certainly need that f \in Cr( - 1, 1), but this condition is not sufficient. If f \in \BbbB r
\infty (w\alpha ,\beta )\cap
\cap rCr( - 1, 1) and r/2 + \alpha , r/2 + \beta \geq 0, then we can only conclude that \omega \varphi
k,r(f
(r), t)\alpha ,\beta ,\infty < \infty
for t > 0. For example, if at least one of r/2 + \alpha and r/2 + \beta is not zero, and f is such that
f (r)(x) := w - 1
\alpha ,\beta (x)\varphi
- r(x), r \in \BbbN 0, then f \in \BbbB r
\infty (w\alpha ,\beta ) \cap Cr( - 1, 1) and \omega \varphi
k,r(f
(r), t)\alpha ,\beta ,\infty \geq 1.
Lemma 2.4. If k \in \BbbN , r \in \BbbN 0, r/2 + \alpha \geq 0, r/2 + \beta \geq 0, and f \in \BbbB r
\infty (w\alpha ,\beta ) \cap Cr( - 1, 1),
then
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow 0
\omega \varphi
k,r(f
(r), t)\alpha ,\beta ,\infty = 0 (2.11)
if and only if
Case 1. r/2 + \alpha > 0 and r/2 + \beta > 0:
\mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow \pm 1
w\alpha ,\beta (x)\varphi
r(x)f (r)(x) = 0. (2.12)
Case 2. r/2 + \alpha > 0 and r/2 + \beta = 0:
\mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow 1
w\alpha ,\beta (x)\varphi
r(x)f (r)(x) = 0, and f (r) \in C[ - 1, 1). (2.13)
Case 3. r/2 + \alpha = 0 and r/2 + \beta > 0:
\mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow - 1
w\alpha ,\beta (x)\varphi
r(x)f (r)(x) = 0, and f (r) \in C( - 1, 1]. (2.14)
Case 4. r/2 + \alpha = 0 and r/2 + \beta = 0:
f (r) \in C[ - 1, 1]. (2.15)
Note that since, for f \in Br
\infty (w\alpha ,\beta ), f (r) may not be defined at \pm 1, when we write f (r) \in
\in C[ - 1, 1), for example, we mean that f (r) can be defined at - 1 so that it becomes continuous
there.
Proof. Since \omega \varphi
k,r(f
(r), t)\alpha ,\beta ,\infty = \omega \varphi
k,0(g, t)r/2+\alpha ,r/2+\beta ,\infty with g := f (r), without loss of
generality, we may assume that r = 0 throughout this proof. Note also that Case 4 is trivial since
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ON MODULI OF SMOOTHNESS WITH JACOBI WEIGHTS 393
\omega \varphi
k,0(f, t)0,0,\infty = \omega k
\varphi (f, t)\infty , the regular DT modulus, tends to 0 as t \rightarrow 0 if and only if f is
uniformly continuous (= continuous) on [ - 1, 1].
We now prove the lemma in Case 2, all other cases being similar.
Given \varepsilon > 0, assume that (2.13) holds, and let \delta = \delta (\varepsilon ) \in (0, 1) be such that
w\alpha ,\beta (x)| f(x)| < 2 - k\varepsilon , x \in [1 - \delta , 1).
Denote
\omega (t) := \omega k(f, t; [ - 1, 1 - \delta /3]),
the regular kth modulus of smoothness of f on the interval [ - 1, 1 - \delta /3], and note that \mathrm{l}\mathrm{i}\mathrm{m}t\rightarrow 0 \omega (t) =
= 0 because of the continuity of f on this interval. Thus, there exists t0 > 0 such that t0 \leq 2\delta /(3k)
and \omega (t0) < \varepsilon /2\alpha , and we fix 0 < h \leq t0.
For x \in \frakD kh, denote Jx := [x - kh\varphi (x)/2, x + kh\varphi (x)/2] \subseteq [ - 1, 1]. If x \leq 1 - 2\delta /3, then
Jx \subseteq [ - 1, 1 - \delta /3]. Hence,
| \scrW \alpha ,\beta
kh (x)\Delta k
h\varphi (x)(f, x)| \leq 2\alpha | \Delta k
h\varphi (x)(f, x)| < \varepsilon . (2.16)
If, on the other hand, x > 1 - 2\delta /3, then Jx \subseteq [1 - \delta , 1]. Hence, for some \theta \in Jx,
| \scrW \alpha ,\beta
kh (x)\Delta k
h\varphi (x)(f, x)| \leq 2k\scrW \alpha ,\beta
kh (x)| f(\theta )| \leq 2kw\alpha ,\beta (\theta )| f(\theta )| < \varepsilon . (2.17)
Combining (2.16) and (2.17), we get (2.11).
Conversely, assume that \alpha > 0, \beta = 0 and (2.11) holds. Observing that \mathrm{l}\mathrm{i}\mathrm{m}t\rightarrow 0 \omega k(f, t; [ - 1, 0]) =
= 0, we conclude that f is uniformly continuous on [ - 1, 0], i.e., f \in C[ - 1, 1). Also, given \varepsilon > 0,
fix 0 < h < 1/(2k) such that \omega \varphi
k,0(f, h)\alpha ,\beta ,\infty < \varepsilon . Let x \in (3/4, 1), and let \theta \in (1/2, x) be such
that \theta + kh\varphi (\theta )/2 = x. Then
| f(x) - \Delta k
h\varphi (\theta )(f, \theta )| \leq (2k - 1)\| f\| C[0,1 - h2/4] =: Ah
which yields
| w\alpha ,\beta (x)f(x)| \leq
w\alpha ,\beta (x)
\scrW \alpha ,\beta
kh (\theta )
| \scrW \alpha ,\beta
kh (\theta )\Delta k
h\varphi (\theta )(f, \theta )| + w\alpha ,\beta (x)Ah \leq
\leq \omega \varphi
k,0(f, h)\alpha ,\beta ,\infty + w\alpha ,\beta (x)Ah.
Hence, \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}x\rightarrow 1 | w\alpha ,\beta (x)f(x)| \leq \varepsilon , and so \mathrm{l}\mathrm{i}\mathrm{m}x\rightarrow 1w\alpha ,\beta (x)(x)f(x) = 0.
Lemma 2.4 is proved.
3. Proof of the upper estimate in Theorem 1.1. We devote this section to proving that
the moduli defined by (1.1) can be estimated from above by the appropriate K -functionals from
Definition 1.4.
First, we need the following lemma.
Lemma 3.1. Let k \in \BbbN , r \in \BbbN 0, r/2+\alpha \geq 0, r/2+\beta \geq 0, and 1 \leq p \leq \infty . If g \in \BbbB r+k
p (w\alpha ,\beta ),
then
\omega \varphi
k,r(g
(r), t)\alpha ,\beta , p \leq ctk
\bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
k+rg(k+r)
\bigm\| \bigm\| \bigm\|
p
. (3.1)
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394 K. A. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
Proof. We follow the lines of the proof of [10] (Lemma 4.1) and rely on the calculations there,
modified to accommodate the additional weight w\alpha ,\beta .
We begin with the well known identity
\Delta k
h(F, x) =
h/2\int
- h/2
. . .
h/2\int
- h/2
F (k)(x+ u1 + . . .+ uk)du1 . . . duk (3.2)
and write
\omega \varphi
k,r(g
(r), t)\alpha ,\beta , p = \mathrm{s}\mathrm{u}\mathrm{p}
0<h\leq t
\bigm\| \bigm\| \bigm\| \scrW r/2+\alpha ,r/2+\beta
kh \Delta k
h\varphi (g
(r), \cdot )
\bigm\| \bigm\| \bigm\|
Lp(\frakD kh)
=
= \mathrm{s}\mathrm{u}\mathrm{p}
0<h\leq t
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \scrW r/2+\alpha ,r/2+\beta
kh
h\varphi /2\int
- h\varphi /2
. . .
h\varphi /2\int
- h\varphi /2
g(k+r)(\cdot + u1 + . . .+ uk)du1 . . . duk
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Lp(\frakD kh)
\leq
\leq \mathrm{s}\mathrm{u}\mathrm{p}
0<h\leq t
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
h\varphi /2\int
- h\varphi /2
. . .
h\varphi /2\int
- h\varphi /2
\bigl(
w\alpha ,\beta \varphi
r| g(k+r)|
\bigr)
(\cdot + u1 + . . .+ uk)du1 . . . duk
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Lp(\frakD kh)
,
where, in the last inequality, we used the fact that r/2 + \alpha \geq 0 and r/2 + \beta \geq 0 implies
\scrW r/2+\alpha ,r/2+\beta
kh (x) \leq w\alpha ,\beta (v)\varphi
r(v), if x - kh\varphi (x)/2 \leq v \leq x+ kh\varphi (x)/2.
By Hölder’s inequality (with 1/p+1/q = 1), for each x \in \frakD kh and | u| \leq (k - 1)h\varphi (x)/2, we have
h\varphi (x)/2\int
- h\varphi (x)/2
\bigl(
w\alpha ,\beta \varphi
r| g(k+r)|
\bigr)
(x+ u+ uk)duk =
x+u+h\varphi (x)/2\int
x+u - h\varphi (x)/2
\bigl(
w\alpha ,\beta \varphi
r| g(k+r)|
\bigr)
(v)dv \leq
\leq \| w\alpha ,\beta \varphi
k+rg(k+r)\| Lp(\scrA (x,u))
\bigm\| \bigm\| \bigm\| \varphi - k
\bigm\| \bigm\| \bigm\|
Lq(\scrA (x,u))
\leq
\leq \scrG \alpha ,\beta p (x; g, k, r)
\bigm\| \bigm\| \bigm\| \varphi - k
\bigm\| \bigm\| \bigm\|
Lq(\scrA (x,u))
,
where
\scrA (x, u) :=
\biggl[
x+ u - h
2
\varphi (x), x+ u+
h
2
\varphi (x)
\biggr]
and
\scrG \alpha ,\beta p (x; g, k, r) :=
\bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
k+rg(k+r)
\bigm\| \bigm\| \bigm\|
Lp[x - kh\varphi (x)/2,x+kh\varphi (x)/2]
.
Thus, the proof is complete, once we show that
I(k, p) \leq chk
\bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
k+rg(k+r)
\bigm\| \bigm\| \bigm\|
p
, (3.3)
where
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ON MODULI OF SMOOTHNESS WITH JACOBI WEIGHTS 395
I(k, p) :=
\bigm\| \bigm\| \bigm\| \scrG \alpha ,\beta p (\cdot ; g, k, r)\scrF q(\cdot , k)
\bigm\| \bigm\| \bigm\|
Lp(\frakD kh)
,
\scrF q(x, k) :=
h\varphi (x)/2\int
- h\varphi (x)/2
. . .
h\varphi (x)/2\int
- h\varphi (x)/2
\bigm\| \bigm\| \bigm\| \varphi - k
\bigm\| \bigm\| \bigm\|
Lq(\scrA (x,u1+...+uk - 1))
du1 . . . duk - 1, if k \geq 2,
and \scrF q(x, 1) :=
\bigm\| \bigm\| \varphi - 1
\bigm\| \bigm\|
Lq(\scrA (x,0))
.
To this end, we write
\| \cdot \| Lp(\frakD kh)
\leq \| \cdot \| Lp(\frakD 2kh)
+ \| \cdot \| Lp((\frakD kh\setminus \frakD 2kh)\cap [0,1]) + \| \cdot \| Lp((\frakD kh\setminus \frakD 2kh)\cap [ - 1,0]) =:
=: I1(p) + I2(p) + I3(p).
In order to estimate I1(p), using (2.9), for x \in \frakD 2kh, we have
\scrF q(x, k) \leq 2k(h\varphi (x))k - 1\varphi - k(x)(h\varphi (x))1/q = 2khk - 1/p\varphi - 1/p(x).
Exactly the same sequence of inequalities as in [10, p. 141, 142] with \varphi k+rg(k+r) there replaced by
w\alpha ,\beta \varphi
k+rg(k+r) yields the estimate
I1(p) \leq chk
\bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
k+rg(k+r)
\bigm\| \bigm\| \bigm\|
p
.
We now estimate I2(p), the estimate of I3(p) being analogous. Denoting
\scrE kh := (\frakD kh \setminus \frakD 2kh) \cap [0, 1]
we note that, since \scrG \alpha ,\beta p (x; g, k, r) \leq
\bigm\| \bigm\| w\alpha ,\beta \varphi
k+rg(k+r)
\bigm\| \bigm\|
p
, x \in \frakD kh, we are done if we show that
\| \scrF q(\cdot , k)\| Lp(\scrE kh) \leq chk. (3.4)
It remains to observe that the estimates\int
\scrE kh
(\scrF q(x, k))
p dx \leq chkp and \mathrm{s}\mathrm{u}\mathrm{p}
x\in \scrE kh
\scrF 1(x, k) \leq chk
which are, respectively, inequalities (4.19) and (4.10) from [10], imply the validity of (3.4).
Lemma 3.1 is proved.
Lemma 3.2. Let k \in \BbbN , r \in \BbbN 0, r/2+\alpha \geq 0, r/2+\beta \geq 0, and 1 \leq p \leq \infty . If f \in \BbbB r
p(w\alpha ,\beta ),
then
\omega \varphi
k,r(f
(r), t)\alpha ,\beta , p \leq cK\varphi
k,r(f
(r), tk)\alpha ,\beta , p, t > 0. (3.5)
Proof. Take any g \in \BbbB r+k
p (w\alpha ,\beta ). Then, by Lemma 2.1, g \in \BbbB r
p(w\alpha ,\beta ), and using Lemmas 2.2
and 3.1 we have
\omega \varphi
k,r(f
(r), t)\alpha ,\beta , p \leq \omega \varphi
k,r(f
(r) - g(r), t)\alpha ,\beta , p + \omega \varphi
k,r(g
(r), t)\alpha ,\beta , p \leq
\leq c
\bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
r
\Bigl(
f (r) - g(r)
\Bigr) \bigm\| \bigm\| \bigm\|
p
+ ctk
\bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
k+rg(k+r)
\bigm\| \bigm\| \bigm\|
p
,
which immediately yields (3.5).
Lemma 3.2 is proved.
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396 K. A. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
4. Equivalence of the moduli and Realization functionals and proof of the lower estimate
in Theorem 1.1. In this section, using some general results for special classes of doubling and A\ast
weights, we prove that, for all 0 < p \leq \infty , the \omega \varphi
k,r moduli are equivalent to certain Realization
functionals. This, in turn, provides lower estimates of \omega \varphi
k,r by means of the appropriate K -functionals,
thus proving the lower estimate in Theorem 1.1. This, of course, is meaningful only for 1 \leq p \leq \infty ,
as we recall that, for 0 < p < 1, the K -functionals may vanish while the moduli do not.
For general definitions of doubling weights, A\ast weights, \scrW (\scrZ ) and \scrW \ast (\scrZ ) see [5, 6]. We only
mentioned that the Jacobi weights with nonnegative exponents belong to all of these classes (see [6]
(Remark 3.3) and [5] (Example 2.7)). We now restate some definitions from [5, 6], adapting them to
the weights w\alpha ,\beta with \alpha , \beta \geq 0, and state corresponding theorems for these weights only.
Let \scrZ 1
A,h := [ - 1, - 1 +Ah2], \scrZ 2
A,h := [1 - Ah2, 1] and \scrI A,h := [ - 1 +Ah2, 1 - Ah2].
The main part weighted modulus of smoothness and the averaged main part weighted modulus
are defined, respectively, as
\Omega k
\varphi (f,A, t)p,w := \mathrm{s}\mathrm{u}\mathrm{p}
0<h\leq t
\bigm\| \bigm\| \bigm\| w(\cdot )\Delta k
h\varphi (\cdot )(f, \cdot ; \scrI A,h)
\bigm\| \bigm\| \bigm\|
Lp(\scrI A,h)
and
\widetilde \Omega k
\varphi (f,A, t)p,w :=
\left( 1
t
t\int
0
\bigm\| \bigm\| \bigm\| w(\cdot )\Delta k
h\varphi (\cdot )(f, \cdot ; \scrI A,h)
\bigm\| \bigm\| \bigm\| p
Lp(\scrI A,h)
dh
\right) 1/p .
The (complete) weighted modulus of smoothness and the (complete) averaged weighted modulus
are defined as
\omega k
\varphi (f,A, t)p,w := \Omega k
\varphi (f,A, t)p,w +
2\sum
j=1
Ek(f,\scrZ j
2A,t)w,p
and
\widetilde \omega k
\varphi (f,A, t)p,w := \widetilde \Omega k
\varphi (f,A, t)p,w +
2\sum
j=1
Ek(f,\scrZ j
2A,t)w,p,
respectively.
The following is an immediate corollary of [5] (Theorem 5.2) in the case 0 < p < \infty and [6]
(Theorem 6.1) if p =\infty .
Theorem 4.1. Let k, \nu 0 \in \BbbN , \nu 0 \geq k, 0 < p \leq \infty , \alpha \geq 0, \beta \geq 0, A > 0, and f \in L\alpha ,\beta
p . Then,
there exists N \in \BbbN depending on k, \nu 0, p, \alpha and \beta , such that for every n \geq N and \vargamma > 0, there is
a polynomial Pn \in \BbbP n satisfying
\| w\alpha ,\beta (f - Pn)\| p \leq c\widetilde \omega k
\varphi (f,A, \vargamma /n)p,w\alpha ,\beta
\leq c\omega k
\varphi (f,A, \vargamma /n)p,w\alpha ,\beta
and
n - \nu
\bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
\nu P (\nu )
n
\bigm\| \bigm\| \bigm\|
p
\leq c\widetilde \omega k
\varphi (f,A, \vargamma /n)p,w\alpha ,\beta
\leq c\omega k
\varphi (f,A, \vargamma /n)p,w\alpha ,\beta
, k \leq \nu \leq \nu 0,
where constants c depend only on k, \nu 0, p, A, \alpha , \beta and \vargamma .
The following theorem is proved in [11].
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ON MODULI OF SMOOTHNESS WITH JACOBI WEIGHTS 397
Theorem 4.2. Let k \in \BbbN , \alpha \geq 0, \beta \geq 0, A > 0, 0 < p \leq \infty , and f \in L\alpha ,\beta
p . Then, for any
0 < t \leq
\sqrt{}
2/A, we have
Ek(f,\scrZ A,t)w\alpha ,\beta ,p \leq c\omega \ast \varphi
k,0(f, t)\alpha ,\beta ,p \leq c\omega \varphi
k,0(f, t)\alpha ,\beta ,p, (4.1)
where the interval \scrZ A,t is either [1 - At2, 1] or [ - 1, - 1 +At2], and c depends only on k, p, \alpha , \beta ,
and A.
In particular, if A = 2 and t = 1, then
Ek(f)w\alpha ,\beta ,p \leq c\omega \ast \varphi
k,0(f, 1)\alpha ,\beta ,p \leq c\omega \varphi
k,0(f, 1)\alpha ,\beta ,p. (4.2)
We now show that the moduli \omega k
\varphi (f,A, t)p,w\alpha ,\beta
and \widetilde \omega k
\varphi (f,A, t)p,w\alpha ,\beta
may be estimated from
above by the moduli \omega \varphi
k,0(f, t)\alpha ,\beta ,p and \omega \ast \varphi
k,0(f, t)\alpha ,\beta ,p, respectively.
Lemma 4.1. Let k \in \BbbN , \alpha \geq 0, \beta \geq 0, A \geq 2k2 and f \in L\alpha ,\beta
p , 0 < p \leq \infty . Then, for
0 < t \leq 1/
\surd
A,
\omega k
\varphi (f,A, t)p,w\alpha ,\beta
\leq c\omega \varphi
k,0(f, t)\alpha ,\beta ,p
and \widetilde \omega k
\varphi (f,A, t)p,w\alpha ,\beta
\leq c\omega \ast \varphi
k,0(f, t)\alpha ,\beta ,p,
where constants c depend only on k, p, \alpha , \beta and A.
Proof. Recall that \scrI A,h = [ - 1 +Ah2, 1 - Ah2] and note that, if A \geq 2k2, then
\scrI A,h \subseteq \frakD 2kh \subset \frakD kh for all h > 0.
Since, by Proposition 2.1, w\alpha ,\beta (x) \sim \scrW \alpha ,\beta
kh (x), x \in \frakD 2kh, we have\bigm\| \bigm\| \bigm\| w\alpha ,\beta (\cdot )\Delta k
h\varphi (\cdot )(f, \cdot ; \scrI A,h)
\bigm\| \bigm\| \bigm\|
Lp(\scrI A,h)
\leq c
\bigm\| \bigm\| \bigm\| \scrW \alpha ,\beta
kh (\cdot )\Delta k
h\varphi (\cdot )(f, \cdot )
\bigm\| \bigm\| \bigm\|
Lp(\frakD kh)
,
so that
\Omega k
\varphi (f,A, t)p,w\alpha ,\beta
\leq c\omega \varphi
k,0(f, t)\alpha ,\beta ,p
and \widetilde \Omega k
\varphi (f,A, t)p,w\alpha ,\beta
\leq c\omega \ast \varphi
k,0(f, t)\alpha ,\beta ,p.
Now, Theorem 4.2 yields that, for 0 < t \leq 1/
\surd
A,
\mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
Ek(f, [1 - 2At2, 1])w\alpha ,\beta ,p, Ek(f, [ - 1, - 1 + 2At2])w\alpha ,\beta ,p
\bigr\}
\leq
\leq c\omega \ast \varphi
k,0(f, t)\alpha ,\beta ,p \leq c\omega \varphi
k,0(f, t)\alpha ,\beta ,p.
Lemma 4.1 is proved.
The following is an immediate corollary of Theorem 4.1 and Lemma 4.1.
Corollary 4.1. Let k \in \BbbN , r \in \BbbN 0, r/2+\alpha \geq 0, r/2+ \beta \geq 0, and f \in \BbbB r
p(w\alpha ,\beta ), 0 < p \leq \infty .
Then, there exists N \in \BbbN depending on k, r, p, \alpha and \beta , such that for every n \geq N and 0 < \vargamma \leq 1,
there is a polynomial Pn \in \BbbP n satisfying\bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
r(f (r) - P (r)
n )
\bigm\| \bigm\| \bigm\|
p
\leq c\omega \ast \varphi
k,r(f
(r), \vargamma /n)\alpha ,\beta ,p \leq c\omega \varphi
k,r(f
(r), \vargamma /n)\alpha ,\beta ,p
and
n - k
\bigm\| \bigm\| \bigm\| w\alpha ,\beta \varphi
k+rP (k+r)
n
\bigm\| \bigm\| \bigm\|
p
\leq c\omega \ast \varphi
k,r(f
(r), \vargamma /n)\alpha ,\beta ,p \leq c\omega \varphi
k,r(f
(r), \vargamma /n)\alpha ,\beta ,p,
where constants c depend only on k, r, p, \alpha , \beta and \vargamma .
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
398 K. A. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
Suppose now that 0 < t \leq 2/k, and n \in \BbbN is such that n \geq N and c1/t \leq n \leq c2/t. Then,
denoting \mu := \mathrm{m}\mathrm{a}\mathrm{x}\{ 1, c2\} , Corollary 4.1 with \vargamma = \mathrm{m}\mathrm{i}\mathrm{n}\{ 1, c1\} implies that
K\varphi
k,r(f
(r), tk)\alpha ,\beta ,p \leq \mu kK\varphi
k,r(f
(r), (t/\mu )k)\alpha ,\beta ,p \leq \mu kR\varphi
k,r(f
(r), n - k)\alpha ,\beta ,p \leq
\leq c\omega \ast \varphi
k,r(f
(r), \vargamma /n)\alpha ,\beta ,p \leq c\omega \ast \varphi
k,r(f
(r), t)\alpha ,\beta ,p \leq c\omega \varphi
k,r(f
(r), t)\alpha ,\beta ,p. (4.3)
Note that (4.3) is valid for all 0 < p \leq \infty . However, we remind the reader that, for 0 < p < 1,
the K -functional may become identically equal to zero.
Together with Lemma 3.2, the sequence of estimates (4.3) immediately yields Theorem 1.1.
We now show that the estimates in Lemma 4.1 may be reversed in some sense, i.e., there exists
0 < \theta \leq 1 such that moduli \omega \varphi
k,0(f, \theta t)\alpha ,\beta ,p and \omega \ast \varphi
k,0(f, \theta t)\alpha ,\beta ,p may be estimated from above,
respectively, by \omega k
\varphi (f,A, t)p,w\alpha ,\beta
and \widetilde \omega k
\varphi (f,A, t)p,w\alpha ,\beta
.
Lemma 4.2. Let k \in \BbbN , \alpha \geq 0, \beta \geq 0, A > 0 and f \in L\alpha ,\beta
p , 0 < p \leq \infty . Then, there exists
0 < \theta \leq 1 depending only on k and A, such that for all 0 < t \leq
\sqrt{}
1/A,
\omega \varphi
k,0(f, \theta t)\alpha ,\beta ,p \leq c\omega k
\varphi (f,A, t)p,w\alpha ,\beta
(4.4)
and
\omega \ast \varphi
k,0(f, \theta t)\alpha ,\beta ,p \leq c\widetilde \omega k
\varphi (f,A, t)p,w\alpha ,\beta
, (4.5)
where constants c depend only on k, p, \alpha , \beta and A.
Proof. Let B := \mathrm{m}\mathrm{a}\mathrm{x}\{ A2, 4k2\} , \theta := \mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
1,
\sqrt{}
A/(kB)
\Bigr\}
, 0 < t \leq
\sqrt{}
1/A and 0 < h \leq \theta t.
Note that h \leq
\sqrt{}
1/B and, if x \in \scrI B,h, then x \pm kh\varphi (x)/2 \in \scrI A,h. Also, \scrI B,h \subset \frakD 2kh, and so
Proposition 2.1 implies that w\alpha ,\beta (x) \sim \scrW \alpha ,\beta
kh (x), for all x \in \scrI B,h. Hence,\bigm\| \bigm\| \bigm\| \scrW \alpha ,\beta
kh (\cdot )\Delta k
h\varphi (\cdot )(f, \cdot )
\bigm\| \bigm\| \bigm\|
Lp(\scrI B,h)
\leq c
\bigm\| \bigm\| \bigm\| w\alpha ,\beta (\cdot )\Delta k
h\varphi (\cdot )(f, \cdot ; \scrI A,h)
\bigm\| \bigm\| \bigm\|
Lp(\scrI A,h)
. (4.6)
Now, let S1 := [0, 1] \cap (\frakD kh \setminus \scrI B,h) . Then, denoting x0 = 1 - Bh2, we have
\widetilde S1 :=
\bigcup
x\in S1
[x - kh\varphi (x)/2, x+ kh\varphi (x)/2] = [x0 - kh\varphi (x0)/2, 1] \subset [1 - 2At2, 1].
It now follows by Remark 2.3 that, for a polynomial of best weighted approximation pk \in \BbbP k to f
on [1 - 2At2, 1], \bigm\| \bigm\| \bigm\| \scrW \alpha ,\beta
kh (\cdot )\Delta k
h\varphi (\cdot )(f, \cdot )
\bigm\| \bigm\| \bigm\|
Lp(S1)
\leq c \| w\alpha ,\beta (f - pk)\| Lp(\widetilde S1)
\leq
\leq cEk(f, [1 - 2At2, 1])w\alpha ,\beta ,p, (4.7)
where we used the fact that any kth difference of pk is identically zero.
Similarly, for S2 := [ - 1, 0] \cap (\frakD kh \setminus \scrI B,h) and
\widetilde S2 :=
\bigcup
x\in S2
[x - kh\varphi (x)/2, x+ kh\varphi (x)/2] \subset [ - 1, - 1 + 2At2],
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
ON MODULI OF SMOOTHNESS WITH JACOBI WEIGHTS 399
we have \bigm\| \bigm\| \bigm\| \scrW \alpha ,\beta
kh (\cdot )\Delta k
h\varphi (\cdot )(f, \cdot )
\bigm\| \bigm\| \bigm\|
Lp(S2)
\leq cEk(f, [ - 1, - 1 + 2At2])w\alpha ,\beta ,p. (4.8)
Therefore, noting that \frakD kh = \scrI B,h \cup S1 \cup S2 and combining (4.6) through (4.8), we have, for all
0 < h \leq \theta t,
\bigm\| \bigm\| \bigm\| \scrW \alpha ,\beta
kh (\cdot )\Delta k
h\varphi (\cdot )(f, \cdot )
\bigm\| \bigm\| \bigm\|
Lp(\frakD kh)
\leq c
\bigm\| \bigm\| \bigm\| w\alpha ,\beta (\cdot )\Delta k
h\varphi (\cdot )(f, \cdot ; \scrI A,h)
\bigm\| \bigm\| \bigm\|
Lp(\scrI A,h)
+ c
2\sum
j=1
Ek(f,\scrZ j
2A,t)w,p.
Estimates (4.4) and (4.5) now follow, respectively, by taking supremum and by integrating with
respect to h over (0, \theta t], and using the fact that \theta \leq 1.
Lemma 4.2 is proved.
Using Lemmas 4.1 and 4.2 we immediately get Theorem 1.2 as a corollary of the following result
that follows from [5] (Corollary 11.2).
Theorem 4.3. Let k \in \BbbN , 0 < p < 1, A > 0, \alpha \geq 0, \beta \geq 0, and f \in L\alpha ,\beta
p . Then there exist
N \in \BbbN depending on k, p, \alpha and \beta , and \vargamma > 0 depending on k, p, A, \alpha and \beta , such that, for any
\vargamma 1 \in (0, \vargamma ], n \geq N, \vargamma 1/n \leq t \leq \vargamma /n, we have
R\varphi
k,0(f, n
- k)\alpha ,\beta ,p \sim \widetilde \omega k
\varphi (f,A, t)p,w\alpha ,\beta
\sim \omega k
\varphi (f,A, t)p,w\alpha ,\beta
.
5. Weighted DT moduli and alternative proof of the lower estimate via \bfitK -functionals. In
this section, we provide an alternative proof, in the case 1 \leq p \leq \infty , of the lower estimate of the
moduli \omega \varphi
k,r(f
(r), t)\alpha ,\beta ,p and \omega \ast \varphi
k,r(f
(r), t)\alpha ,\beta ,p by appropriate K -functionals, using certain weighted
DT moduli.
We denote the kth forward and the kth backward differences by
- \rightarrow
\Delta k
h(f, x) := \Delta k
h(f, x+ kh/2)
and
\leftarrow -
\Delta k
h(f, x) := \Delta k
h(f, x - kh/2), respectively.
Adapting the weighted DT moduli which were defined in [1] ((8.2.10)) for a weight w on
D := [ - 1, 1], we set for f \in Lw,p,
\omega k
\varphi (f, t)w,p := \mathrm{s}\mathrm{u}\mathrm{p}
0<h\leq t
\bigm\| \bigm\| \bigm\| w(\cdot )\Delta k
h\varphi (f, \cdot )
\bigm\| \bigm\| \bigm\|
Lp[ - 1+t\ast ,1 - t\ast ]
+
+ \mathrm{s}\mathrm{u}\mathrm{p}
0<h\leq t\ast
\bigm\| \bigm\| \bigm\| w(\cdot ) - \rightarrow \Delta k
h(f, \cdot )
\bigm\| \bigm\| \bigm\|
Lp[ - 1, - 1+12t\ast ]
+
+ \mathrm{s}\mathrm{u}\mathrm{p}
0<h\leq t\ast
\bigm\| \bigm\| \bigm\| w(\cdot )\leftarrow - \Delta k
h(f, \cdot )
\bigm\| \bigm\| \bigm\|
Lp[1 - 12t\ast ,1]
, (5.1)
where t\ast := 2k2t2. The first term on the right in the above equation is called the main-part modulus
and denoted by \Omega k
\varphi (f, t)w,p. Obviously, we have \Omega k
\varphi (f, t)w,p \leq \omega k
\varphi (f, t)w,p.
Next, the weighted K -functional was defined in [1, p. 55] ((6.1.1)) as
Kk,\varphi (f, t
k)w,p := \mathrm{i}\mathrm{n}\mathrm{f}
g\in \BbbB k
p(w)
\{ \| w(f - g)\| p + tk\| w\varphi kg(k)\| p\} ,
and we note that
Kk,\varphi (f, t
k)w\alpha ,\beta ,p = K\varphi
k,0(f, t
k)\alpha ,\beta ,p.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
400 K. A. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
It was shown in [1] (Theorem 6.1.1) that, given an appropriate weight w (all Jacobi weights with
nonnegative exponents are included), the weighted K -functional is equivalent to the weighted DT
modulus of f. Namely, by [1] (Theorem 6.1.1), for 1 \leq p \leq \infty ,
M - 1\omega k
\varphi (f, t)w,p \leq Kk,\varphi (f, t
k)w,p \leq M\omega k
\varphi (f, t)w,p, 0 < t \leq t0,
where t0 is some sufficiently small constant. Hence, in particular, if \alpha , \beta \geq 0, then
\omega k
\varphi (f, t)w\alpha ,\beta ,p \sim Kk,\varphi (f, t
k)w\alpha ,\beta ,p, 0 < t \leq t0. (5.2)
Note that, if \alpha < 0 or \beta < 0, then there are functions f in L\alpha ,\beta
p for which \omega k
\varphi (f, \delta )w\alpha ,\beta ,p =\infty .
Indeed, the following example was given in [4] (see also [1, p. 56] (Remark 6.1.2)) and, in fact,
it was the starting point for our counterexample in Theorem 2.1. Suppose that 1 \leq p < \infty and
that \delta > 0 is fixed. If f(x) := (x + 1 - \varepsilon ) - \beta - 1/p\chi [ - 1+\varepsilon , - 1+2\varepsilon ](x) with \beta < 0 and 0 < \varepsilon <
< t\ast , then \| w\alpha ,\beta f\| p \leq c(\alpha , \beta , p) (and so f \in L\alpha ,\beta
p ), \| w\alpha ,\beta (\cdot )f(\cdot + \varepsilon )\| Lp[ - 1, - 1+12t\ast ] = \infty , and
\| w\alpha ,\beta (\cdot )f(\cdot + i\varepsilon )\| p = 0, 2 \leq i \leq k, and therefore
\mathrm{s}\mathrm{u}\mathrm{p}
0<h\leq t\ast
\bigm\| \bigm\| \bigm\| w\alpha ,\beta (\cdot )
- \rightarrow
\Delta k
h(f, \cdot )
\bigm\| \bigm\| \bigm\|
Lp[ - 1, - 1+12t\ast ]
\geq
\bigm\| \bigm\| \bigm\| w\alpha ,\beta (\cdot )
- \rightarrow
\Delta k
\varepsilon (f, \cdot )
\bigm\| \bigm\| \bigm\|
Lp[ - 1, - 1+12t\ast ]
=
= \| w\alpha ,\beta (\cdot ) [f(\cdot ) - kf(\cdot + \varepsilon )]\| Lp[ - 1, - 1+12t\ast ] =\infty .
Also, if f \in L\alpha ,\beta
p then (choosing g \equiv 0) we have Kk,\varphi (f, t
k)w\alpha ,\beta ,p \leq \| w\alpha ,\beta f\| p <\infty . Hence, (5.2)
is not valid if \alpha < 0 or \beta < 0 (see also Theorem 2.1 with r = 0).
An equivalent averaged weighted DT modulus
\omega \ast k
\varphi (f, t)w,p :=
\left( 1
t
t\int
0
\int 1 - t\ast
- 1+t\ast
| w(x)\Delta k
\tau \varphi (x)(f, x)|
p dx d\tau
\right) 1/p+
+
\left( 1
t\ast
t\ast \int
0
- 1+At\ast \int
- 1
| w(x)
- \rightarrow
\Delta k
u(f, x)| p dx du
\right) 1/p+
+
\left( 1
t\ast
t\ast \int
0
1\int
1 - At\ast
| w(x)
\leftarrow -
\Delta k
u(f, x)| p dx du
\right) 1/p , (5.3)
where 1 \leq p < \infty , t\ast := 2k2t2, and A is some sufficiently large absolute constant, was defined
in [1] ((6.1.9)). For p = \infty , set \omega \ast k
\varphi (f, t)w,\infty := \omega k
\varphi (f, t)w,\infty . It was shown in [1, p. 57] that,
for an appropriate weight w (again, all Jacobi weights with nonnegative exponents are included),
1 \leq p \leq \infty and sufficiently small t0 > 0,
Kk,\varphi (f, t
k)w,p \leq c\omega \ast k
\varphi (f, t)w,p, 0 < t \leq t0. (5.4)
We now provide an alternative proof of the inverse estimate to (3.5) independent of the results in
Section 4. First, we need the following lemma.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
ON MODULI OF SMOOTHNESS WITH JACOBI WEIGHTS 401
Lemma 5.1. Let k \in \BbbN , r \in \BbbN 0, r/2 + \alpha \geq 0, r/2 + \beta \geq 0, 1 \leq p <\infty , and f \in \BbbB r
p(w\alpha ,\beta ).
Then
\omega \ast k
\varphi (f (r), t)w\alpha ,\beta \varphi r,p \leq c(k, r, \alpha , \beta )\omega \ast \varphi
k,r(f
(r), c(k)t)\alpha ,\beta , p, 0 < t \leq c(k).
Proof. The proof of this lemma is very similar to that of Lemma 6.1 in [10], but we still provide
all details here for completeness. The three terms in the definition (5.3) are to be estimated separately,
but the second and third are similar, so we will estimate the first two. Since \omega \ast k
\varphi (f (r), t)w\alpha ,\beta \varphi r,p =
= \omega \ast k
\varphi (g, t)wr/2+\alpha ,r/2+\beta ,p and \omega \ast \varphi
k,r(f
(r), t)\alpha ,\beta ,p = \omega \ast \varphi
k,0(g, t)r/2+\alpha ,r/2+\beta ,p with g := f (r), without loss
of generality, we may assume that r = 0 throughout this proof.
Note that t\ast = 2k2t2 implies that [ - 1+ t\ast , 1 - t\ast ] \subset \frakD 2kt \subset \frakD 2k\tau , 0 \leq \tau \leq t, so that by (2.10)
we have
1
t
t\int
0
1 - t\ast \int
- 1+t\ast
| w\alpha ,\beta (x)\Delta
k
\tau \varphi (x)(f, x)|
p dx d\tau \leq
\leq 2(\alpha +\beta )p
t
t\int
0
\int
\frakD 2k\tau
| \scrW \alpha ,\beta
k\tau (x)\Delta k
\tau \varphi (x)(f, x)|
p dx d\tau \leq
\leq 2(\alpha +\beta )p\omega \ast \varphi
k,0(f, t)
p
\alpha ,\beta , p .
In order to estimate the second term we follow the proof of [10] (Lemma 6.1) and assume that
t \leq (2k
\sqrt{}
A+ k/2) - 1. Then
1
t\ast
t\ast \int
0
- 1+At\ast \int
- 1
| w\alpha ,\beta (x)
- \rightarrow
\Delta k
u(f, x)| p dx du =
=
1
t\ast
t\ast \int
0
- 1+At\ast \int
- 1
| w\alpha ,\beta (x)\Delta
k
u(f, x+ ku/2)| p dx du \leq
\leq 1
t\ast
t\ast \int
0
- 1+(A+k/2)t\ast \int
- 1+ku/2
| w\alpha ,\beta (y - ku/2)\Delta k
u(f, y)| p dy du \leq
\leq 1
t\ast
- 1+(A+k/2)t\ast \int
- 1
2(y+1)/k\int
0
| w\alpha ,\beta (y - ku/2)\Delta k
u(f, y)| p du dy =
=
1
t\ast
- 1+(A+k/2)t\ast \int
- 1
2(y+1)/(k\varphi (y))\int
0
\varphi (y)| w\alpha ,\beta (y - kh\varphi (y)/2)\Delta k
h\varphi (y)(f, y)|
p dh dy \leq
\leq c
1
t\ast
- 1+(A+k/2)t\ast \int
- 1
2(y+1)/(k\varphi (y))\int
0
\varphi (y)| \scrW \alpha ,\beta
kh (y)\Delta k
h\varphi (y)(f, y)|
p dh dy \leq
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
402 K. A. KOPOTUN, D. LEVIATAN, I. A. SHEVCHUK
\leq c
1\surd
t\ast
- 1+(A+k/2)t\ast \int
- 1
2(y+1)/(k\varphi (y))\int
0
| \scrW \alpha ,\beta
kh (y)\Delta k
h\varphi (y)(f, y)|
p dh dy \leq
\leq c
1\surd
t\ast
c
\surd
t\ast \int
0
\int
\frakD kh\cap [ - 1, - 1+(A+k/2)t\ast ]
| \scrW \alpha ,\beta
kh (y)\Delta k
h\varphi (y)(f, y)|
p dy dh \leq
\leq c\omega \ast \varphi
k,0(f, c(k)t)
p
\alpha ,\beta ,p,
where for the third inequality we used the fact that, for y \leq - 1/2 and 0 \leq h \leq 2(y + 1)/(k\varphi (y)),
1 - y + kh\varphi (y)/2 \leq 2 (1 - y - kh\varphi (y)/2) ,
and so
w\alpha ,\beta (y - kh\varphi (y)/2) \leq 2\alpha \scrW \alpha ,\beta
kh (y).
Lemma 5.1 is proved.
A similar proof yields (see [10], Lemma 6.2) an analogous result in the case p =\infty .
Lemma 5.2. Let k \in \BbbN , r \in \BbbN 0, r/2 + \alpha \geq 0, r/2 + \beta \geq 0, and f \in \BbbB r
\infty (w\alpha ,\beta ). Then
\omega k
\varphi (f
(r), t)w\alpha ,\beta \varphi r,\infty \leq c(k, r, \alpha , \beta )\omega \varphi
k,r
\bigl(
f (r), c(k)t
\bigr)
\alpha ,\beta ,\infty , 0 < t \leq c(k).
We are now ready to prove the inverse of the estimate (3.5).
Lemma 5.3. Let k \in \BbbN , r \in \BbbN 0, r/2+\alpha \geq 0, r/2+\beta \geq 0, and 1 \leq p \leq \infty . If f \in \BbbB r
p(w\alpha ,\beta ),
then
K\varphi
k,r(f
(r), tk)\alpha ,\beta , p \leq c\omega \ast \varphi
k,r(f
(r), t)\alpha ,\beta , p \leq c\omega \varphi
k,r(f
(r), t)\alpha ,\beta , p, 0 < t \leq 2/k. (5.5)
Proof. Combining (5.4) with the weight w = w\alpha ,\beta \varphi
r with Lemmas 5.1 and 5.2, we obtain, for
1 \leq p \leq \infty ,
K\varphi
k,r(f
(r), tk)\alpha ,\beta , p = Kk,\varphi (f
(r), tk)w\alpha ,\beta \varphi r, p \leq
\leq c\omega \ast k
\varphi (f (r), t)w\alpha ,\beta \varphi r, p \leq c\omega \ast \varphi
k,r(f
(r), c(k)t)\alpha ,\beta , p , 0 < t \leq c .
Hence, we have
K\varphi
k,r(f
(r), tk)\alpha ,\beta , p \leq c\omega \ast \varphi
k,r(f
(r), c1t)\alpha ,\beta , p, 0 < t \leq c2, (5.6)
where c1 and c2 are some positive constants that may depend only on k.
Suppose now that 0 < t \leq 2/k. Then, denoting \mu := \mathrm{m}\mathrm{a}\mathrm{x}\{ 1, c1, 2/(kc2)\} and using (5.6) we
obtain
K\varphi
k,r(f
(r), tk)\alpha ,\beta , p \leq \mu kK\varphi
k,r(f
(r), (t/\mu )k)\alpha ,\beta , p \leq
\leq c\omega \ast \varphi
k,r(f
(r), c1t/\mu )\alpha ,\beta ,p \leq c\omega \ast \varphi
k,r(f
(r), t)\alpha ,\beta , p ,
which is the first inequality in (5.5). Finally, the second inequality in (5.5) follows from (1.4).
Lemma 5.3 is proved.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
ON MODULI OF SMOOTHNESS WITH JACOBI WEIGHTS 403
References
1. Ditzian Z., Totik V. Moduli of smoothness // Springer Ser. Comput. Math. – New York: Springer-Verlag, 1987. –
Vol. 9.
2. Dzyadyk V. K., Shevchuk I. A. Theory of uniform approximation of functions by polynomials. – Berlin: Walter de
Gruyter, 2008.
3. Gonska H. H., Leviatan D., Shevchuk I. A., Wenz H.-J. Interpolatory pointwise estimates for polynomial approxima-
tion // Constr. Approxim. – 2000. – 16, № 4. – P. 603 – 629.
4. Kopotun K. A. Weighted moduli of smoothness of k-monotone functions and applications // J. Approxim. Theory. –
2015. – 192. – P. 102 – 131.
5. Kopotun K. A. Polynomial approximation with doubling weights having finitely many zeros and singularities // J.
Approxim. Theory. – 2015. – 198. – P. 24 – 62.
6. Kopotun K. A. Uniform polynomial approximation with A\ast weights having finitely many zeros // J. Math. Anal. and
Appl. – 2016. – 435, № 1. – P. 677 – 700.
7. Kopotun K. A., Leviatan D., Prymak A. V., Shevchuk I. A. Uniform and pointwise shape preserving approximation by
algebraic polynomials // Surv. Approxim. Theory. – 2011. – 6. – P. 24 – 74.
8. Kopotun K. A., Leviatan D., Shevchuk I. A. Are the degrees of the best (co)convex and unconstrained polynomial
approximations the same? II // Ukr. Math. J. – 2010. – 62, № 3. – P. 420 – 440.
9. Kopotun K. A., Leviatan D., Shevchuk I. A. New moduli of smoothness // Publ. Inst. Math. Serb. Acad. Sci. and Arts
Belgrade. – 2014. – 96(110). – P. 169 – 180.
10. Kopotun K. A., Leviatan D., Shevchuk I. A. New moduli of smoothness: weighted DT moduli revisited and applied //
Constr. Approxim. – 2015. – 42. – P. 129 – 159.
11. Kopotun K.A., Leviatan D., Shevchuk I. A. On weighted approximation with Jacobi weights, https://arxiv.org/abs/
1710.05059
Received 29.11.17
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 3
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| resource_txt_mv | umjimathkievua/b0/a946219a47004f58d5a5d58ada0421b0.pdf |
| spelling | umjimathkievua-article-15632019-12-05T09:18:29Z On moduli of smoothness with Jacobi weights Про модулi гладкостi з вагами Якобi Kopotun, K. A. Leviatan, D. Shevchuk, I. A. Копотун, К. А. Левіатан, Д. Шевчук, І. О. We introduce the moduli of smoothness with Jacobi weights $(1 x)\alpha (1+x)\beta$ for functions in the Jacobi weighted spaces $L_p[ 1, 1],\; 0 < p \leq \infty $. These moduli are used to characterize the smoothness of (the derivatives of) functions in the weighted spaces $L_p$. If $1 \leq p \leq \infty$, then these moduli are equivalent to certain weighted $K$-functionals (and so they are equivalent to certain weighted Ditzian – Totik moduli of smoothness for these $p$), while for $0 < p < 1$ they are equivalent to certain “Realization functionals”. Введено модулi гладкостi з вагами Якобi $(1 x)\alpha (1 + x)\beta$ для функцiй, що належать ваговим просторам Якобi $L_p[ 1, 1],\; 0 < p \leq \infty $. Цi модулi використовуються, щоб охарактеризувати гладкiсть функцiй та їх похiдних у вагових просторах $L_p$. При $1 \leq p \leq \infty $ цi модулi еквiвалентнi деяким ваговим K-функцiоналам (таким чином, еквiвалентнi деяким ваговим модулям гладкостi Дiцiана – Тотiка для цих $p$). Водночас при $0 < p < 1$ цi модулi еквiвалентнi деяким „функцiоналам реалiзацiй”. Institute of Mathematics, NAS of Ukraine 2018-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1563 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 3 (2018); 379-403 Український математичний журнал; Том 70 № 3 (2018); 379-403 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1563/545 Copyright (c) 2018 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 moduli of smoothness with Jacobi weights |
| title | On moduli of smoothness with Jacobi weights |
| title_alt | Про модулi гладкостi з вагами Якобi |
| title_full | On moduli of smoothness with Jacobi weights |
| title_fullStr | On moduli of smoothness with Jacobi weights |
| title_full_unstemmed | On moduli of smoothness with Jacobi weights |
| title_short | On moduli of smoothness with Jacobi weights |
| title_sort | on moduli of smoothness with jacobi weights |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1563 |
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