On On the approximation of functions by Jacobi – Dunkl expansion in the weighted space $\mathbb{L}_{2}^{(\alpha,\beta)}$
UDC 517.5 We prove some new estimates useful in applications  for the approximation of certain classes of functions characterized by the generalized continuity modulus from the space $\mathbb{L}_{2}^{(\alpha,\beta)}$ by partial sums of the Jacobi – Dunkl series. For t...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512315866087424 |
|---|---|
| author | Tyr, O. Daher, R. Tyr, O. Daher, R. |
| author_facet | Tyr, O. Daher, R. Tyr, O. Daher, R. |
| author_sort | Tyr, O. |
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| datestamp_date | 2022-12-17T13:38:56Z |
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UDC 517.5
We prove some new estimates useful in applications  for the approximation of certain classes of functions characterized by the generalized continuity modulus from the space $\mathbb{L}_{2}^{(\alpha,\beta)}$ by partial sums of the Jacobi – Dunkl series. For this purpose, we use the generalized Jacobi – Dunkl translation operator obtained  by Vinogradov in the monograph [Theory of approximation of functions of real variable, Fizmatgiz, Moscow (1960) (in Russian)]. |
| doi_str_mv | 10.37863/umzh.v74i10.6275 |
| first_indexed | 2026-03-24T03:26:50Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v74i10.6275
UDC 517.5
O. Tyr1, R. Daher (Univ. Hassan II, Casablanca, Morocco)
ON THE APPROXIMATION OF FUNCTIONS
BY JACOBI – DUNKL EXPANSION IN THE WEIGHTED SPACE \BbbL (\bfitalpha ,\bfitbeta )
\bftwo
ПРО НАБЛИЖЕННЯ ФУНКЦIЙ ЗА ДОПОМОГОЮ РОЗКЛАДIВ
ЯКОБI – ДАНКЛА У ВАГОВОМУ ПРОСТОРI \BbbL (\bfitalpha ,\bfitbeta )
\bftwo
We prove some new estimates useful in applications for the approximation of certain classes of functions characterized by
the generalized continuity modulus from the space \BbbL (\alpha ,\beta )
2 by partial sums of the Jacobi – Dunkl series. For this purpose, we
use the generalized Jacobi – Dunkl translation operator obtained by Vinogradov in the monograph [Theory of approximation
of functions of real variable, Fizmatgiz, Moscow (1960) (in Russian)].
Доведено деякi новi оцiнки, кориснi в застосуваннях, для наближень певних класiв функцiй, що характеризуються
узагальненим модулем неперервностi з простору \BbbL (\alpha ,\beta )
2 , частковими сумами рядiв Якобi – Данкла. З цiєю метою
використано узагальнений оператор трансляцiї Якобi – Данкла, що був отриманий Виноградовим у монографiї
[Theory of approximation of functions of real variable. Fizmatgiz, Moscow (1960) (in Russian)].
1. Introduction. It is well-known that many problems for partial differential equations are reduced
to a power series expansion of the desired solution in terms of special functions or orthogonal
polynomials (such as Laguerre, Hermite, Jacobi, etc. polynomials). In particular, this is associated
with the separation of variables as applied to problems in mathematical physics (see, e.g., [10, 11]).
In [2], Abilov et al. proved two useful estimates for the Fourier transform in the space of
square integrable functions on certain classes of functions characterized by the generalized continuity
modulus, using a translation operator. In this paper, we also discuss this subject. More specially, we
prove some estimates (similar to those proved in [2]) in certain classes of functions characterized by
a generalized continuity modulus and connected with the discrete Jacobi – Dunkl transform associated
with the Jacobi – Dunkl operator defined on \BbbT = [ - \pi /2, \pi /2] by
\Lambda \alpha ,\beta f(\theta ) :=
d
d\theta
f(\theta ) +
\scrA \prime
\alpha ,\beta (\theta )
\scrA \alpha ,\beta (\theta )
f(\theta ) - f( - \theta )
2
, f \in \scrC 1
\Bigl( \Bigr]
- \pi
2
,
\pi
2
\Bigl[ \Bigr)
,
where
\alpha \geq \beta \geq - 1
2
, \alpha \not = - 1
2
.
This paper is organized as follows. In Section 2, we state some basic notions and results from the
discrete harmonic analysis associated with the Jacobi – Dunkl transform that will be needed through-
out this paper. Some estimates are proved in Section 3.
2. Preliminaries. In this section, we will recall some properties of Jacobi and Jacobi – Dunkl
polynomials, we develop some results from the discrete harmonic analysis related to the differential-
difference operator \Lambda \alpha ,\beta . Further details can be found in [3 – 5, 7, 8, 12]. In the following, we fix
parameters \alpha and \beta subject to the constraints
1 Corresponding author, e-mail: otywac@hotmail.fr.
c\bigcirc O. TYR, R. DAHER, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10 1427
1428 O. TYR, R. DAHER
\alpha \geq \beta \geq - 1
2
, \alpha \not = - 1
2
,
and set
\rho = \alpha + \beta + 1.
The Gauss hypergeometric function 2F1(a, b; c; z) is defined by
2F1(a, b; c; z) :=
+\infty \sum
n=0
(a)n(b)n
(c)nn!
zn, | z| < 1,
where a, b, c, z \in \BbbC with c /\in \BbbZ - and (a)n is the Pochhammer symbol given by
(a)n :=
\left\{ a(a+ 1) . . . (a+ n - 1) if n \in \BbbN \ast ,
1 if n = 0,
where \BbbN \ast = \{ 1, 2, . . .\} .
The Jacobi polynomials \varphi (\alpha ,\beta )
n (\theta ), n \in \BbbN , \theta \in \BbbT , are defined by
\varphi (\alpha ,\beta )
n (\theta ) := \scrR (\alpha ,\beta )
n (\mathrm{c}\mathrm{o}\mathrm{s}(2\theta )) = 2F1
\bigl(
- n, n+ \rho ;\alpha + 1; \mathrm{s}\mathrm{i}\mathrm{n}2 \theta
\bigr)
with \scrR (\alpha ,\beta )
n (x), n \in \BbbN , is the normalized Jacobi polynomial of degree n such that \scrR (\alpha ,\beta )
n (1) = 1.
Note that, for all n \in \BbbN , we have \bigm| \bigm| \bigm| \varphi (\alpha ,\beta )
n (\theta )
\bigm| \bigm| \bigm| \leq 1 \forall \theta \in \BbbT (1)
and
\varphi (\alpha ,\beta )
n ( - \theta ) = \varphi (\alpha ,\beta )
n (\theta ) \forall \theta \in \BbbT . (2)
The Jacobi operator \Delta \alpha ,\beta defined on \scrC 2
\Bigl( \Bigr]
0,
\pi
2
\Bigl[ \Bigr)
is given by
\Delta \alpha ,\beta :=
d2
d\theta 2
+
\scrA \prime
\alpha ,\beta
\scrA \alpha ,\beta
d
d\theta
with
\scrA \alpha ,\beta (\theta ) :=
\left\{ 22\rho (\mathrm{s}\mathrm{i}\mathrm{n} | \theta | )2\alpha +1(\mathrm{c}\mathrm{o}\mathrm{s} \theta )2\beta +1 if \theta \in
\Bigr]
- \pi
2
,
\pi
2
\Bigl[
\setminus \{ 0\} ,
0 if \theta = 0.
For all n \in \BbbN , \varphi (\alpha ,\beta )
n is the unique even \scrC \infty -solution on
\Bigr]
- \pi
2
,
\pi
2
\Bigl[
of the differential equation
\Delta \alpha ,\beta f = - \lambda 2nf,
f(0) = 1,
f \prime (0) = 0,
where
\lambda n = \lambda (\alpha ,\beta )n := 2\mathrm{s}\mathrm{g}\mathrm{n}(n)
\sqrt{}
| n| (| n| + \rho ), n \in \BbbZ .
The Jacobi function \varphi (\alpha ,\beta )
n , n \in \BbbN , satisfies the following inequalities.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
ON THE APPROXIMATION OF FUNCTIONS BY JACOBI – DUNKL EXPANSION . . . 1429
Lemma 1. The following inequalities are valid for Jacobi functions \varphi (\alpha ,\beta )
n :
a) for \theta \in (0, \pi /4], we have
1 - \varphi (\alpha ,\beta )
n (\theta ) \leq c1\lambda
2
n\theta
2, (3)
b) for every \gamma > 0, there is a number c2 = c2(\gamma , \alpha , \beta ) > 0 such that, for all n and \theta with
\gamma < n\theta <
\pi n
4
, we obtain \bigm| \bigm| \bigm| \varphi (\alpha ,\beta )
n (\theta )
\bigm| \bigm| \bigm| \leq c2(n\theta )
- \alpha - 1/2. (4)
Proof. See [9] (Proposition 3.5 and Lemma 3.1).
The Jacobi – Dunkl operator \Lambda \alpha ,\beta is defined by
\Lambda \alpha ,\beta f(\theta ) :=
d
d\theta
f(\theta ) +
\scrA \prime
\alpha ,\beta (\theta )
\scrA \alpha ,\beta (\theta )
f(\theta ) - f( - \theta )
2
, f \in \scrC 1
\Bigl( \Bigr]
- \pi
2
,
\pi
2
\Bigl[ \Bigr)
,
with
\scrA \prime
\alpha ,\beta (\theta )
\scrA \alpha ,\beta (\theta )
= (2\alpha + 1) \mathrm{c}\mathrm{o}\mathrm{t} \theta + (2\beta + 1) \mathrm{t}\mathrm{a}\mathrm{n} \theta , \theta \in
\Bigr] \pi
2
,
\pi
2
\Bigl[
\setminus \{ 0\} .
From [7], for all n \in \BbbZ , the differential-difference equation
\Lambda \alpha ,\beta f(\theta ) = i\lambda nf(\theta ), n \in \BbbZ ,
f(0) = 1,
admits a unique \scrC \infty -solution \psi (\alpha ,\beta )
n (\theta ) on
\Bigr]
- \pi
2
,
\pi
2
\Bigl[
. It is related to the Jacobi polynomial and to its
derivative by
\psi (\alpha ,\beta )
n (\theta ) :=
\left\{ \varphi
(\alpha ,\beta )
| n| (\theta ) - i
\lambda n
d
d\theta
\varphi
(\alpha ,\beta )
| n| (\theta ) if n \in \BbbZ \ast ,
1 if n = 0.
We note that, for all n \in \BbbZ and \theta \in \BbbT , we have
\psi
(\alpha ,\beta )
- n (\theta ) = \psi (\alpha ,\beta )
n ( - \theta ) = \psi
(\alpha ,\beta )
n (\theta ), (5)\bigm| \bigm| \bigm| \psi (\alpha ,\beta )
n (\theta )
\bigm| \bigm| \bigm| \leq 1,
and, for all f and g such that
\int \pi
2
- \pi
2
\Lambda \alpha ,\beta f(\theta )g(\theta )\scrA \alpha ,\beta (\theta )d\theta exists, we obtain
\pi
2\int
- \pi
2
\Lambda \alpha ,\beta f(\theta )g(\theta )\scrA \alpha ,\beta (\theta )d\theta = -
\pi
2\int
- \pi
2
f(\theta )\Lambda \alpha ,\beta g(\theta )\scrA \alpha ,\beta (\theta )d\theta . (6)
For all n, p \in \BbbZ , we have the orthogonality formula given by (see [7])
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1430 O. TYR, R. DAHER
\pi
2\int
- \pi
2
\psi (\alpha ,\beta )
n (\theta )\psi
(\alpha ,\beta )
p (\theta )\scrA \alpha ,\beta (\theta )d\theta =
\Bigl(
w(\alpha ,\beta )
n
\Bigr) - 1
\delta n,p, (7)
where
w(\alpha ,\beta )
n =
\left(
\pi
2\int
- \pi
2
\bigm| \bigm| \bigm| \psi (\alpha ,\beta )
n (\theta )
\bigm| \bigm| \bigm| 2\scrA \alpha ,\beta (\theta )d\theta
\right)
- 1
, w
(\alpha ,\beta )
0 =
\Gamma (\rho + 1)
22\rho \Gamma (\alpha + 1)\Gamma (\beta + 1)
and
w(\alpha ,\beta )
n =
(2| n| + \rho )\Gamma (\alpha + | n| + 1)\Gamma (\rho + | n| )
22\rho +1(\Gamma (\alpha + 1))2\Gamma (| n| + 1)\Gamma (\beta + | n| + 1)
\forall n \in \BbbZ \ast .
We obtain the following asymptotic equality as n\rightarrow +\infty :
w(\alpha ,\beta )
n \asymp | n| 2\alpha +1
22\rho (\Gamma (\alpha + 1))2
.
By using the relation (see [7])
d
d\theta
\varphi
(\alpha ,\beta )
| n| (\theta ) = - \lambda 2n
4(\alpha + 1)
\mathrm{s}\mathrm{i}\mathrm{n}(2\theta )\varphi
(\alpha +1,\beta +1)
| n| - 1 (\theta ),
the function \psi (\alpha ,\beta )
n can be written in the form
\psi (\alpha ,\beta )
n (\theta ) = \varphi
(\alpha ,\beta )
| n| (\theta ) + i
\lambda n
4(\alpha + 1)
\mathrm{s}\mathrm{i}\mathrm{n}(2\theta )\varphi
(\alpha +1,\beta +1)
| n| - 1 (\theta ). (8)
Let \BbbL (\alpha ,\beta )
2 denote the space of square integrable functions f(\theta ) on the closed interval \BbbT with the
weight function \scrA \alpha ,\beta (\theta ) and the norm
\| f\| =
\sqrt{}
\pi
2\int
- \pi
2
| f(\theta )| 2\scrA \alpha ,\beta (\theta )d\theta .
We define the weighted spaces l2(\BbbZ ) := l2
\Bigl(
\BbbZ , w(\alpha ,\beta )
n
\Bigr)
by
l2(\BbbZ ) =
\Biggl\{
(fn)n\in \BbbZ : \BbbZ - \rightarrow \BbbC :
+\infty \sum
n= - \infty
| fn| 2w(\alpha ,\beta )
n < +\infty
\Biggr\}
.
The Jacobi – Dunkl expansion of a function f \in \BbbL (\alpha ,\beta )
2 is defined by (see [6, 7])
f(\theta ) =
+\infty \sum
n= - \infty
\scrF f(n)\psi (\alpha ,\beta )
n (\theta )w(\alpha ,\beta )
n \forall \theta \in \BbbT , (9)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
ON THE APPROXIMATION OF FUNCTIONS BY JACOBI – DUNKL EXPANSION . . . 1431
where
\scrF f(n) :=
\pi
2\int
- \pi
2
f(\theta )\psi
(\alpha ,\beta )
n (\theta )\scrA \alpha ,\beta (\theta )d\theta \forall n \in \BbbZ .
The sequence \{ \scrF f(n), n \in \BbbZ \} is called the discrete Jacobi – Dunkl transform of f. For n \in \BbbN , we
denote the partial sum of (9) by
\scrS fn(\theta ) :=
n\sum
k= - n
\scrF f(k)\psi (\alpha ,\beta )
k (\theta )w
(\alpha ,\beta )
k \forall \theta \in \BbbT .
From (7), we get, for all f \in \BbbL (\alpha ,\beta )
2 (see [6]),
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow +\infty
\bigm\| \bigm\| \bigm\| \scrS fn - f
\bigm\| \bigm\| \bigm\| = 0.
We state some properties of the discrete Jacobi – Dunkl transform \scrF (see [7]).
Theorem 1 (Plancherel formula). If f \in \BbbL (\alpha ,\beta )
2 , then \scrF f belongs to l2(\BbbZ ) and we have
\| f\| =
\sqrt{} +\infty \sum
n= - \infty
| \scrF f(n)| 2w(\alpha ,\beta )
n . (10)
Proof. See [7] (Theorem 3.4).
The generalized Jacobi – Dunkl translation operator is defined for f \in \BbbL (\alpha ,\beta )
2 and \theta , h \in \BbbT by
\scrT hf(\theta ) :=
\left\{
\int \pi
2
- \pi
2
f(\varphi )W (h, \theta , \varphi )\scrA \alpha ,\beta (\varphi )d\varphi if h, \theta \in G\alpha ,\beta ,
f(\theta + h) if h /\in G\alpha ,\beta or \theta /\in G\alpha ,\beta ,
where
G\alpha ,\beta :=
\left\{
\BbbR \setminus \{ n\pi \} n\in \BbbZ if \alpha > \beta \geq - 1
2
,
\BbbR \setminus
\Bigl\{ n\pi
2
\Bigr\}
n\in \BbbZ
if \alpha = \beta > - 1
2
,
\emptyset if \alpha = \beta = - 1
2
,
and W (h, \theta , \varphi ) is a certain function satisfies the following properties (see [12]):
W (h, \theta , \varphi ) =W (\theta , h, \varphi ),
W (h, \theta , - \varphi ) =W ( - h, - \theta , \varphi ),
W (h, \theta , \varphi ) =W (h, - \varphi , - \theta ).
In particular, the product formula
\scrT h\psi (\alpha ,\beta )
n (\theta ) = \psi (\alpha ,\beta )
n (h)\psi (\alpha ,\beta )
n (\theta ). (11)
holds. Some properties of generalized Jacobi – Dunkl translation operator are fulfilled.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1432 O. TYR, R. DAHER
Theorem 2. If f \in \BbbL (\alpha ,\beta )
2 , then \scrT hf \in \BbbL (\alpha ,\beta )
2 and we have
\| \scrT hf\| \leq \| f\| \forall h \in \BbbT .
Proof. See [12] (Theorem 3).
Proposition 1. Let f \in \BbbL (\alpha ,\beta )
2 and n \in \BbbZ . Then
\scrF (\scrT hf)(n) = \psi (\alpha ,\beta )
n (h)\scrF f(n) \forall h \in \BbbT .
Proof. See [12] (Remark 4).
For every f \in \BbbL (\alpha ,\beta )
2 , we define the differences \Delta m
h f of order m, m = 1, 2, . . . , with step
h > 0, 0 < h < \pi /2, by
\Delta 1
hf(\theta ) = \Delta hf(\theta ) := \scrT hf(\theta ) + \scrT - hf(\theta ) - 2f(\theta ),
\Delta m
h f(\theta ) = \Delta h
\bigl(
\Delta m - 1
h f(\theta )
\bigr)
for m \geq 2.
Also, we can write that
\Delta m
h f(\theta ) =
\Bigl(
\scrT h + \scrT - h - 2I\BbbL 2
\Bigr) m
f(\theta ),
where I\BbbL 2 is the identity operator in \BbbL (\alpha ,\beta )
2 .
The generalized modulus of continuity of a function f \in \BbbL (\alpha ,\beta )
2 is defined by
\omega m(f, \delta ) = \mathrm{s}\mathrm{u}\mathrm{p}
0<h\leq \delta
\| \Delta m
h f\| , \delta > 0.
Let W r,m
2,\psi (\Lambda \alpha ,\beta ), r = 0, 1, . . . , denote the class of functions f \in \BbbL (\alpha ,\beta )
2 that have generalized
derivatives satisfying the estimate
\omega m
\bigl(
\Lambda r\alpha ,\beta f, \delta
\bigr)
= O(\psi (\delta m)), \delta \rightarrow 0,
where \psi (.) is any nonnegative function given on [0,+\infty ), \psi (0) = 0 and
\Lambda 0
\alpha ,\beta f = f,
\Lambda r\alpha ,\beta f = \Lambda \alpha ,\beta ,
\Lambda r - 1
\alpha ,\beta f, r = 1, 2, . . . ,
i.e.,
W r,m
2,\psi (\Lambda \alpha ,\beta ) =
\Bigl\{
f \in \BbbL (\alpha ,\beta )
2 : \Lambda r\alpha ,\beta f \in \BbbL (\alpha ,\beta )
2 and \omega m
\bigl(
\Lambda r\alpha ,\beta f, \delta
\bigr)
= O(\psi (\delta m)), \delta \rightarrow 0
\Bigr\}
.
3. Main results. Taking into account what was said in the previous section, for some classes of
functions characterized by the generalized modulus of continuity, we can prove two estimates for the
serie
EN (f) =
\sqrt{} \sum
| n| \geq N
| \scrF f(n)| 2w(\alpha ,\beta )
n ,
which are useful in applications. To prove the main results, we shall need some preliminary results.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
ON THE APPROXIMATION OF FUNCTIONS BY JACOBI – DUNKL EXPANSION . . . 1433
Lemma 2. For f \in \BbbL (\alpha ,\beta )
2 , we get
\scrF (\Lambda \alpha ,\beta f)(n) = i\lambda n\scrF f(n)
for all n \in \BbbZ .
Proof. Since
\lambda - n = - \lambda n \forall n \in \BbbZ ,
it follows from this and (5) that
\Lambda \alpha ,\beta \psi
(\alpha ,\beta )
n (\theta ) = \Lambda \alpha ,\beta \psi
(\alpha ,\beta )
- n (\theta ) = i\lambda - n\psi
(\alpha ,\beta )
- n (\theta ) = - i\lambda n\psi (\alpha ,\beta )
n (\theta ).
Therefore, thanks to the formula (6), we conclude that
\scrF (\Lambda \alpha ,\beta f)(n) =
\pi
2\int
- \pi
2
\Lambda \alpha ,\beta f(\theta )\psi
(\alpha ,\beta )
n (\theta )\scrA \alpha ,\beta (\theta )d\theta =
= -
\pi
2\int
- \pi
2
f(\theta )\Lambda \alpha ,\beta (\psi
(\alpha ,\beta )
n (\theta ))\scrA \alpha ,\beta (\theta )d\theta =
=
\pi
2\int
- \pi
2
i\lambda nf(\theta )\psi
(\alpha ,\beta )
n (\theta )\scrA \alpha ,\beta (\theta )d\theta = i\lambda n\scrF f(n).
Lemma 2 is proved.
Remark 1. From Lemma 2, we can see that, for all f \in W r,m
2,\psi (\Lambda \alpha ,\beta ),
\scrF
\bigl(
\Lambda r\alpha ,\beta f
\bigr)
(n) = (i\lambda n)
r\scrF f(n) \forall n \in \BbbZ
for all r = 0, 1, 2, . . . ,m.
Lemma 3. Let \theta \in \BbbT . If f \in \BbbL (\alpha ,\beta )
2 with
f(\theta ) =
+\infty \sum
n= - \infty
\scrF f(n)\psi (\alpha ,\beta )
n (\theta )w(\alpha ,\beta )
n ,
then
\scrT hf(\theta ) =
+\infty \sum
n= - \infty
\scrF f(n)\psi (\alpha ,\beta )
n (h)\psi (\alpha ,\beta )
n (\theta )w(\alpha ,\beta )
n .
Proof. By product formula (11) of \scrT h, we have
\scrT h\psi (\alpha ,\beta )
n (\theta ) = \psi (\alpha ,\beta )
n (h)\psi (\alpha ,\beta )
n (\theta ).
Thus, for any polynomial
\scrQ N (\theta ) =
N\sum
n= - N
\scrF f(n)\psi (\alpha ,\beta )
n (\theta )w(\alpha ,\beta )
n ,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1434 O. TYR, R. DAHER
since \scrT h is linear, we obtain
\scrT h\scrQ N (\theta ) =
N\sum
n= - N
\scrF f(n)\psi (\alpha ,\beta )
n (h)\psi (\alpha ,\beta )
n (\theta )w(\alpha ,\beta )
n . (12)
By using the fact that \scrT h is a linear bounded operator in \BbbL (\alpha ,\beta )
2 and the set of all polynomials \scrQ N (\theta )
is everywhere dense in \BbbL (\alpha ,\beta )
2 , passage to the limit in (12) gives the desired equality.
Lemma 3 is proved.
Lemma 4. Let f \in W r,m
2,\psi (\Lambda \alpha ,\beta ) and 0 < h < \pi /2. Then, for all n \in \BbbZ , we have
\bigm\| \bigm\| \Delta m
h
\bigl(
\Lambda r\alpha ,\beta f
\bigr) \bigm\| \bigm\| 2 = 22m
+\infty \sum
n= - \infty
\lambda 2rn
\bigm| \bigm| \bigm| 1 - \varphi
(\alpha ,\beta )
| n| (h)
\bigm| \bigm| \bigm| 2m| \scrF f(n)| 2w(\alpha ,\beta )
n ,
where m = 0, 1, 2, . . . and r = 0, 1, 2, . . . ,m.
Proof. Take into account the result of Lemma 3, we get
\Delta hf(\theta ) = \scrT hf(\theta ) + \scrT - hf(\theta ) - 2f(\theta ) =
=
+\infty \sum
n= - \infty
\Bigl(
\psi (\alpha ,\beta )
n (h) + \psi (\alpha ,\beta )
n ( - h) - 2
\Bigr)
\scrF f(n)\psi (\alpha ,\beta )
n (\theta )w(\alpha ,\beta )
n .
Since (see (8))
\psi (\alpha ,\beta )
n (h) = \varphi
(\alpha ,\beta )
| n| (h) + i
\lambda n
4(\alpha + 1)
\mathrm{s}\mathrm{i}\mathrm{n}(2h)\varphi
(\alpha +1,\beta +1)
| n| - 1 (h),
\psi (\alpha ,\beta )
n ( - h) = \varphi
(\alpha ,\beta )
| n| ( - h) - i
\lambda n
4(\alpha + 1)
\mathrm{s}\mathrm{i}\mathrm{n}(2h)\varphi
(\alpha +1,\beta +1)
| n| - 1 ( - h),
by formula (2), we have
\Delta hf(\theta ) = 2
+\infty \sum
n= - \infty
\Bigl(
\varphi
(\alpha ,\beta )
| n| (h) - 1
\Bigr)
\scrF f(n)\psi (\alpha ,\beta )
n (\theta )w(\alpha ,\beta )
n .
Using the proof of recurrence for m, we obtain
\Delta m
h f(\theta ) = 2m
+\infty \sum
n= - \infty
\Bigl(
\varphi
(\alpha ,\beta )
| n| (h) - 1
\Bigr) m
\scrF f(n)\psi (\alpha ,\beta )
n (\theta )w(\alpha ,\beta )
n .
Remark 1 gives
\Delta m
h (\Lambda
r
\alpha ,\beta f)(\theta ) = ir2m
+\infty \sum
n= - \infty
\lambda rn
\Bigl(
\varphi
(\alpha ,\beta )
| n| (h) - 1
\Bigr) m
\scrF f(n)\psi (\alpha ,\beta )
n (\theta )w(\alpha ,\beta )
n .
By appealing the Plancherel formula (10), we get
\bigm\| \bigm\| \Delta m
h
\bigl(
\Lambda r\alpha ,\beta f
\bigr) \bigm\| \bigm\| 2 = 22m
+\infty \sum
n= - \infty
\lambda 2rn
\bigm| \bigm| \bigm| 1 - \varphi
(\alpha ,\beta )
| n| (h)
\bigm| \bigm| \bigm| 2m| \scrF f(n)| 2w(\alpha ,\beta )
n .
Lemma 4 is proved.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
ON THE APPROXIMATION OF FUNCTIONS BY JACOBI – DUNKL EXPANSION . . . 1435
Theorem 3. For functions f \in \BbbL (\alpha ,\beta )
2 in the class W r,m
2,\psi (\Lambda \alpha ,\beta ), there exists a fixed constant
c > 0 such that, for all N > 0, we have
EN (f) = O(\lambda - rN \psi [(c/N)m])
as N \rightarrow \infty , where r = 0, 1, 2, . . . , m = 1, 2, . . . , and \psi (.) is any nonnegative function given on
[0,+\infty ).
Proof. Let f \in W r,m
2,\psi (\Lambda \alpha ,\beta ), by the Hölder inequality for sums, we obtain
E2
N (f) -
\sum
| n| \geq N
\varphi
(\alpha ,\beta )
| n| (h)| \scrF f(n)| 2wn =
\sum
| n| \geq N
\Bigl(
1 - \varphi
(\alpha ,\beta )
| n| (h)
\Bigr)
| \scrF f(n)| 2wn =
=
\sum
| n| \geq N
\biggl(
| \scrF f(n)| 2 -
1
mw
1 - 1
2m
n
\biggr) \biggl( \Bigl(
1 - \varphi
(\alpha ,\beta )
| n| (h)
\Bigr)
| \scrF f(n)|
1
mw
1
2m
n
\biggr)
\leq
\leq
\left( \sum
| n| \geq N
| \scrF f(n)| 2wn
\right) 2m - 1
2m
\left( \sum
| n| \geq N
\Bigl(
1 - \varphi
(\alpha ,\beta )
| n| (h)
\Bigr) 2m
| \scrF f(n)| 2wn
\right) 1
2m
=
= (EN (f))
2m - 1
m
\left( \sum
| n| \geq N
\Bigl(
1 - \varphi
(\alpha ,\beta )
| n| (h)
\Bigr) 2m
| \scrF f(n)| 2wn
\right) 1
2m
.
Since
\lambda 2n \geq \lambda 2N for all | n| \geq N,
we conclude that
E2
N (f) -
\sum
| n| \geq N
\varphi
(\alpha ,\beta )
| n| (h)| \scrF f(n)| 2wn \leq
\leq (EN (f))
2m - 1
m
\left( \lambda - 2r
N
\sum
| n| \geq N
\lambda 2rn
\Bigl(
1 - \varphi
(\alpha ,\beta )
| n| (h)
\Bigr) 2m
| \scrF f(n)| 2wn
\right) 1
2m
\leq
\leq (EN (f))
2m - 1
m
\left( \lambda - 2r
N 22m
\sum
| n| \geq N
\lambda 2rn (1 - \varphi
(\alpha ,\beta )
| n| (h))2m| \scrF f(n)| 2wn
\right) 1
2m
.
From Lemma 4, we have
22m
\sum
| n| \geq N
\lambda 2rn
\Bigl(
1 - \varphi
(\alpha ,\beta )
| n| (h)
\Bigr) 2m
| \scrF f(n)| 2wn \leq
\bigm\| \bigm\| \Delta m
h
\bigl(
\Lambda r\alpha ,\beta f
\bigr) \bigm\| \bigm\| 2.
Thus,
E2
N (f) \leq
\sum
| n| \geq N
\varphi
(\alpha ,\beta )
| n| (h)| \scrF f(n)| 2wn + (EN (f))
2m - 1
m \lambda
- r
m
N
\bigm\| \bigm\| \Delta m
h
\bigl(
\Lambda r\alpha ,\beta f
\bigr) \bigm\| \bigm\| 1
m . (13)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1436 O. TYR, R. DAHER
From (4), we get \sum
| n| \geq N
\varphi
(\alpha ,\beta )
| n| (h)| \scrF f(n)| 2wn \leq c2(Nh)
- \alpha - 1/2E2
N (f).
For f \in W r,m
2,\psi (\Lambda \alpha ,\beta ), there exists a constant C > 0 such that\bigm\| \bigm\| \Delta m
h
\bigl(
\Lambda r\alpha ,\beta f
\bigr) \bigm\| \bigm\| \leq C\psi (hm).
Choose a constant c3 such that the number c4 = 1 - c2c
- \alpha - 1/2
3 is positive. Setting h = c3/N in the
inequality (13), we have
c4E
2
N (f) \leq (EN (f))
2m - 1
m \lambda
- r
m
N C
1
m (\psi [(c3/N)m])
1
m .
By raising both sides to the power m and simplifying by (EN (f))
2m - 1, we finally obtain
cm4 EN (f) \leq C\lambda - rN \psi [(c3/N)m]
for all N > 0.
Hence, the theorem is proved with c = c3.
Theorem 4. Let \phi (t) = t\nu . Then
f \in W r,m
2,\psi (\Lambda \alpha ,\beta )
is equivalent to
EN (f) = O
\bigl(
N - r - m\nu \bigr) ,
where r = 0, 1, 2, . . . , m = 1, 2, . . . , and 0 < \nu < 2.
Proof. Assume that f \in W r,m
2,\psi (\Lambda \alpha ,\beta ), by using the fact that
\lambda N = 2
\sqrt{}
N(N + \rho ) \geq 2N.
Then, from this and according to the Theorem 3, we conclude that
EN (f) = O
\bigl(
N - r - m\nu \bigr) .
This shows us this implication.
We prove necessity. Let
EN (f) = O
\bigl(
N - r - m\nu \bigr) ,
i.e., \sum
| n| \geq N
| \scrF f(n)| 2w(\alpha ,\beta )
n = O
\bigl(
N - 2r - 2m\nu
\bigr)
. (14)
It is easy to show that there exists a function f \in \BbbL (\alpha ,\beta )
2 such that \Lambda r\alpha ,\beta f \in \BbbL (\alpha ,\beta )
2 and
\Lambda r\alpha ,\beta f(\theta ) = ir
+\infty \sum
n= - \infty
\lambda rn\scrF f(n)\psi (\alpha ,\beta )
n (\theta )w(\alpha ,\beta )
n .
From the formula above and Plancherel identity (10), we have
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
ON THE APPROXIMATION OF FUNCTIONS BY JACOBI – DUNKL EXPANSION . . . 1437
\bigm\| \bigm\| \Delta m
h
\bigl(
\Lambda r\alpha ,\beta f
\bigr) \bigm\| \bigm\| 2 = 22m
+\infty \sum
n= - \infty
\lambda 2rn
\bigm| \bigm| \bigm| 1 - \varphi
(\alpha ,\beta )
| n| (h)
\bigm| \bigm| \bigm| 2m| \scrF f(n)| 2w(\alpha ,\beta )
n .
This sum is divided into two
\| \Delta m
h (\Lambda
r
\alpha ,\beta f)\| 2 = \scrI 1 + \scrI 2,
where
\scrI 1 =
\sum
| n| <N
22m\lambda 2rn | 1 - \varphi
(\alpha ,\beta )
| n| (h)| 2m| \scrF f(n)| 2w(\alpha ,\beta )
n
and
\scrI 2 =
\sum
| n| \geq N
22m\lambda 2rn | 1 - \varphi
(\alpha ,\beta )
| n| (h)| 2m| \scrF f(n)| 2w(\alpha ,\beta )
n
with N =
\bigl[
h - 1
\bigr]
is the integer part of h - 1.
Let us now estimate each of them, we estimate \scrI 2, it follows from (1) that
\scrI 2 \leq 24m
\sum
| n| \geq N
\lambda 2rn | \scrF f(n)| 2w(\alpha ,\beta )
n .
Note that
\lambda 2n = 4n2
\biggl(
1 +
\rho
| n|
\biggr)
\leq 4n2(1 + \rho ) for all | n| \geq 1, n \in \BbbZ . (15)
It follows from this that
\scrI 2 \leq 24m(4\rho + 4)r
\sum
| n| \geq N
n2r| \scrF f(n)| 2w(\alpha ,\beta )
n =
= c5
+\infty \sum
j=0
\sum
N+j\leq | n| \leq N+j+1
n2r| \scrF f(n)| 2w(\alpha ,\beta )
n \leq
\leq c5
+\infty \sum
j=0
(N + j + 1)2r
\sum
N+j\leq | n| \leq N+j+1
| \scrF f(n)| 2w(\alpha ,\beta )
n =
= c5
+\infty \sum
j=0
aj(\scrV j - \scrV j+1),
where aj = (N + j + 1)2r and \scrV j =
\sum
| n| \geq N+j
| \scrF f(n)| 2w(\alpha ,\beta )
n . Furthermore, for all integers
M \geq 1, the summation by parts gives
M\sum
j=0
aj(\scrV j - \scrV j+1) = a0\scrV 0 - aM\scrV M+1 +
M\sum
j=1
\scrV j(aj - aj - 1) \leq
\leq a0\scrV 0 +
M\sum
j=1
\scrV j(aj - aj - 1).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1438 O. TYR, R. DAHER
Moreover, by the finite increments theorem, we have
aj - aj - 1 \leq 2r(N + j + 1)2r - 1.
On the other hand, by (14), there exists c6 > 0 such that, for all N > 0,
E2
N (f) \leq c6N
- 2r - 2m\nu .
For N \geq 1, we obtain
M\sum
j=0
aj(\scrV j - \scrV j+1) \leq a0\scrV 0 +
M\sum
j=1
\scrV j(aj - aj - 1) \leq
\leq c6
\biggl(
1 +
1
N
\biggr) 2r
N - 2m\nu + 2rc6
M\sum
j=1
\biggl(
1 +
1
N + j
\biggr) 2r - 1
(N + j) - 1 - 2m\nu \leq
\leq c62
2rN - 2m\nu + 22rrc6
M\sum
j=1
(N + j) - 1 - 2m\nu .
Finally, by the integral comparison test, we get
M\sum
j=1
(N + j) - 1 - 2m\nu \leq
+\infty \sum
\mu =N+1
\mu - 1 - 2m\nu \leq
+\infty \int
N
t - 1 - 2m\nu dt =
1
2m\nu
N - 2m\nu .
Letting M \rightarrow +\infty , we see that, for r \geq 0 and m, \nu > 0, there exists a constant c7 such that, for all
N \geq 1,
\scrI 2 \leq c7N
- 2m\nu .
Consequently, for all h > 0, we have
\scrI 2 \leq c7h
2m\nu . (16)
Now, we estimate \scrI 1. From formulae (3) and (15), we obtain
\scrI 1 \leq 22mc2m1 h4m
\sum
| n| <N
\lambda 2r+4m
n | \scrF f(n)| 2w(\alpha ,\beta )
n \leq
\leq c8h
4m
\sum
| n| <N
n2r+4m| \scrF f(n)| 2w(\alpha ,\beta )
n \leq
\leq c8h
4m
N - 1\sum
j=0
\sum
j\leq | n| \leq j+1
n2r+4m| \scrF f(n)| 2w(\alpha ,\beta )
n \leq
\leq c8h
4m
N - 1\sum
j=0
(j + 1)2r+4m
\sum
j\leq | n| \leq j+1
| \scrF f(n)| 2w(\alpha ,\beta )
n =
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
ON THE APPROXIMATION OF FUNCTIONS BY JACOBI – DUNKL EXPANSION . . . 1439
= c8h
4m
N - 1\sum
j=0
aj(\scrV j - \scrV j+1),
where aj = (j + 1)2r+4m and \scrV j =
\sum
| n| \geq j
| \scrF f(n)| 2w(\alpha ,\beta )
n .
Using a summation by parts and proceeding as with \scrI 2 and the fact that \scrV j \leq c6j
- 2r - 2m\nu by
hypothesis, we get
\scrI 1 \leq c8h
4m
N - 1\sum
j=0
aj(\scrV j - \scrV j+1) \leq c8h
4m
\left( a0\scrV 0 +
N - 1\sum
j=1
\scrV j(aj - aj - 1)
\right) \leq
\leq c8h
4m
\left( \scrV 0 + c6(2r + 4m)
N - 1\sum
j=1
(j + 1)2r+4m - 1j - 2r - 2m\nu
\right) .
From the inequality j + 1 \leq 2j, we conclude that
\scrI 1 \leq c8h
4m
\left( \scrV 0 + c9
N - 1\sum
j=1
j4m - 2m\nu - 1
\right) .
As a consequence of a series comparison, we have the inequality
\mu
N - 1\sum
j=1
j\mu - 1 \leq N\mu for \mu > 0 and N \geq 2.
If \mu = 4m - 2m\nu > 0 for \nu < 2, then we obtain
\scrI 1 \leq c8h
4m
\bigl(
\scrV 0 + c10N
4m - 2m\nu
\bigr)
\leq c8h
4m
\bigl(
\scrV 0 + c10h
2m\nu - 4m
\bigr)
,
since N \leq 1/h.
If h is sufficiently small, then \scrV 0 \leq c10h
2m\nu - 4m. Then we have
\scrI 1 \leq c11h
2m\nu . (17)
Combining the estimates (16) and (17) for \scrI 1 and \scrI 2 gives\bigm\| \bigm\| \Delta m
h
\bigl(
\Lambda r\alpha ,\beta f
\bigr) \bigm\| \bigm\| = O(hm\nu ).
Consequently,
\omega m
\bigl(
\Lambda r\alpha ,\beta f, \delta
\bigr)
= O(\delta m\nu ) = O(\psi (\delta m)).
Therefore, the necessity is proved and the proof of the theorem is completed.
References
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(2008)).
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Received 20.08.20
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
|
| id | umjimathkievua-article-6275 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:26:50Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a6/08f8cea0873e846a902abead600561a6.pdf |
| spelling | umjimathkievua-article-62752022-12-17T13:38:56Z On On the approximation of functions by Jacobi – Dunkl expansion in the weighted space $\mathbb{L}_{2}^{(\alpha,\beta)}$ On the approximation of functions by Jacobi – Dunkl expansion in the weighted space $\mathbb{L}_{2}^{(\alpha,\beta)}$ Tyr, O. Daher, R. Tyr, O. Daher, R. JACOBI – DUNKL EXPANSION UDC 517.5 We prove some new estimates useful in applications&nbsp; for the approximation of certain classes of functions characterized by the generalized continuity modulus from the space $\mathbb{L}_{2}^{(\alpha,\beta)}$ by partial sums of the Jacobi – Dunkl series.&nbsp;For this purpose, we use the generalized Jacobi – Dunkl translation operator obtained&nbsp; by Vinogradov in the monograph [Theory of approximation of functions of real variable, Fizmatgiz, Moscow (1960) (in Russian)]. УДК 517.5 Про наближення функцій за допомогою розкладів Якобі – Данкла у ваговому просторі $\mathbb{L}_{2}^{(\alpha,\beta)}$ Доведено деякі нові оцінки, корисні в застосуваннях, для наближень певних класів функцій, що характеризуються узагальненим модулем неперервності з простору $\mathbb{L}_{2}^{(\alpha,\beta)},$ частковими сумами рядів Якобі – Данкла.&nbsp;З цією метою використано узагальнений оператор трансляції Якобі – Данкла, що був отриманий Виноградовим у монографії [Theory of approximation of functions of real variable. Fizmatgiz, Moscow (1960) (in Russian)].&nbsp; Institute of Mathematics, NAS of Ukraine 2022-11-27 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6275 10.37863/umzh.v74i10.6275 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 10 (2022); 1427 - 1440 Український математичний журнал; Том 74 № 10 (2022); 1427 - 1440 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6275/9318 Copyright (c) 2022 Othman Tyr |
| spellingShingle | Tyr, O. Daher, R. Tyr, O. Daher, R. On On the approximation of functions by Jacobi – Dunkl expansion in the weighted space $\mathbb{L}_{2}^{(\alpha,\beta)}$ |
| title | On On the approximation of functions by Jacobi – Dunkl expansion in the weighted space $\mathbb{L}_{2}^{(\alpha,\beta)}$ |
| title_alt | On the approximation of functions by Jacobi – Dunkl expansion in the weighted space $\mathbb{L}_{2}^{(\alpha,\beta)}$ |
| title_full | On On the approximation of functions by Jacobi – Dunkl expansion in the weighted space $\mathbb{L}_{2}^{(\alpha,\beta)}$ |
| title_fullStr | On On the approximation of functions by Jacobi – Dunkl expansion in the weighted space $\mathbb{L}_{2}^{(\alpha,\beta)}$ |
| title_full_unstemmed | On On the approximation of functions by Jacobi – Dunkl expansion in the weighted space $\mathbb{L}_{2}^{(\alpha,\beta)}$ |
| title_short | On On the approximation of functions by Jacobi – Dunkl expansion in the weighted space $\mathbb{L}_{2}^{(\alpha,\beta)}$ |
| title_sort | on on the approximation of functions by jacobi – dunkl expansion in the weighted space $\mathbb{l}_{2}^{(\alpha,\beta)}$ |
| topic_facet | JACOBI – DUNKL EXPANSION |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6275 |
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