Gen. Math. Notes, Vol. 26, No. 2, February 2015, pp.1-10
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2015
www.i-csrs.org
Available free online at http://www.geman.in
Inverse Theorems of Approximation Theory
in L2(R+, J α,β (x)dx)
R. Daher1 and S. El Ouadih2
1,2
Departement of Mathematics, Faculty of Sciences
Aı̈n Chock, University, Hassan II, Casablanca, Marocco
1
E-mail: rjdaher024@gmail.com
2
E-mail: salahwadih@gmail.com
(Received: 2-7-14 / Accepted: 21-11-14)
Abstract
In this paper, we prove analogues of some inverse theorems of Stechkin ,
for the Jacobi harmonic analysis, using the function with bounded spectrum.
Keywords: Generalized continuity modulus, Function with bounded spectrum, Best approximation.
1
Introduction
Yet by the year 1912, S. Bernstein obtained the estimate inverse to Jakson’s
inequality in the space of continuous functions for some special cases [2], later
S.B.Stechkin [4], M.Timan [7], etc, proved such inverse estimation , including
the case of the space Lp , 1 < p < ∞.
In this paper, we obtain the estimate inverse to Jakson’s inequality in the space
L2 (R+ , J α,β (x)dx) (see [1], theorem 4.2) ,where the modulus of smoothness is
constructed on the basis of the Jacobi generalized translation.
2
2
R. Daher et al.
The Jacobi Transform and its Basic Properties
Let α ≥
−1
,
2
α>β≥
L :=
−1
,
2
ρ = α + β + 1 and let
d2
d
.
+
((2α
+
1)
coth
x
+
(2β
+
1)
tanh
x)
dx2
dx
(α,β)
be the Jacobi differential operator and denote by ϕλ (x) (λ, x ∈ R+ ) the
(α,β)
Jacobi function of order (α, β), the function ϕλ (x) satisfier the differential
equation
Lϕ + (λ2 + ρ2 )ϕ = 0,
0
with the initial conditions ϕ(0) = 1 and ϕ (0) = 0
(α,β)
Lemma 2.1 The following inequalities are valid for Jacobi functions ϕλ
(α,β)
a) | ϕλ (x) | ≤ 1.
(α,β)
b) 1 − ϕλ (x) ≤ x2 (λ2 + ρ2 ).
c) There exists a positive constant c1 such that if λx > 1 , then
(α,β)
1 − ϕλ
(x) ≥ c1 .
Proof: Analogue (see[1], lemmas 3.1-3.2-3.3)
Consider the Hilbert space L2(α,β) (R+ ) = L2 (R+ , J α,β (x)dx) with the norm
12
+∞
Z
2
kf k2 =
| f (x) | J
α,β
(x)dx
,
0
where
J α,β (x) = (2 sinh x)2α+1 (2 cosh x)2β+1 .
The Jacobi transform of a function f ∈ L2(α,β) (R+ ) is defined by
J
α,β
+∞
Z
(f )(λ) =
(α,β)
f (x)ϕλ
(x)J α,β (x)dx.
0
The inversion formula is
Z
f (x) =
+∞
(α,β)
Jα,β (f )(λ)ϕλ
(x)dµ(λ).
0
where dµ(λ) :=
1
2π
| C(λ) |−2 dλ and the C-function C(λ) is defined by
C(λ) =
2ρ Γ(iλ)Γ( 21 (1 + iλ))
.
Γ( 21 (ρ + iλ))Γ( 12 (ρ + iλ) − β)
(x)
Inverse Theorems of Approximation Theory...
3
The Plancherel formula for Jacobi transform (see [8]) is written as
k f kL2 (R+ ,J α,β (x)dx) = kJα,β (f )kL2 (R+ ,dµ(λ)) .
(1)
We note the important property of the Bessel transform
Jα,β (Lf )(λ) = −(λ2 + ρ2 )Jα,β (f )(λ).
(2)
The generalized translation operator was defined by Flented-Jensen and Koornwinder [6] given by
Z +∞
h
f (z)K(x, h, z)J α,β (z)dz.
T f (x) =
0
where the kernel K is explicitly known (see [8]).
In[3], we have
(α,β)
Jα,β (T h f )(λ) = ϕλ
(h)Jα,β (f )(λ), h > 0.
(3)
In [6], we have
kT h f k2 ≤ kf k2 .
(4)
A function f ∈ L2(α,β) (R+ ) is called a function with bounded spectrum of order
ν > 0 if Jα,β (f )(λ) = 0 for λ > ν. The set of all such functions is denoted by
Iν .
Lemma 2.2 For every function f ∈ Iν , ν ≥ 1 and any number m ∈ N we
have
kLm f k2 ≤ c2 ν 2m kf k2 .
where c2 = (1 + ρ2 )m is a constant.
Proof: (see corollary 1.6 in [1])
The finite differences of the first and higher orders are defined as follows:
∆h f (x) = T h f (x) − f (x) = (T h − I)f (x).
where I is the identity operator in L2(α,β) (R+ ), and
h
k
∆kh f (x) = ∆h (∆k−1
h f (x)) = (T − I) f (x).
The generalized continuity modulus of order k in L2(α,β) (R+ ) is defined as follows
ωk (f, δ)2 := sup k∆kh f k2 , δ > 0, f ∈ L2(α,β) (R+ ).
0<h≤δ
4
R. Daher et al.
The best approximation of a function L2(α,β) (R+ ) by functions in Iν is the
quantity
Eν (f )2 := inf kf − gk2 .
g∈Iν
Let
be the Sobolev space of order m ∈ N∗ constructed from the differential operator L, that is,
m
W2,(α,β)
m
W2,(α,β)
= {f ∈ L2(α,β) (R+ ) : Lj f ∈ L2(α,β) (R+ ), j = 1, 2, ..., m}.
where Lj f = L(Lj−1 f ), and L0 f = f .
3
Main Results
In this section we give the main results of this paper
Lemma 3.1 The modulus of smoothness ωk (f, t)2 has the following properties.
i) ωk (f + g, t)2 ≤ ωk (f, t)2 + ωk (g, t)2 .
ii) ωk (f, t)2 ≤ 2k kf k2 .
k
, then
iii) If f ∈ W2,(α,β)
ωk (f, t)2 ≤ t2k kLk f k2 .
Proof:
Prperty 1 follow from the definition of ωk (f, t)2 .
Prperty 2 follow from the fact that kT h f k2 ≤ kf k2 .
Assume that h ∈ (0, t]. From formulas (1), (2) and (3), we have
k α,β
k ∆kh f kL2 (R+ ,J α,β (x)dx) = k(1 − ϕα,β
(f )(λ)kL2 (R+ ,dµ(λ))
λ (h)) J
k
2
2 k α,β
k L f kL2 (R+ ,J α,β (x)dx) = k(λ + ρ ) J
(f )(λ)kL2 (R+ ,dµ(λ)) .
(5)
(6)
Formula (5) implies the equality
k ∆kh f kL2 (R+ ,J α,β (x)dx) = h2k k
k
(1 − ϕα,β
λ (h))
(λ2 + ρ2 )k Jα,β (f )kL2 (R+ ,dµ(λ)) .
h2k (λ2 + ρ2 )k
From lemma (2.1), we obtain
k ∆kh f kL2 (R+ ,J α,β (x)dx) ≤ h2k k(λ2 + ρ2 )k Jα,β (f )kL2 (R+ ,dµ(λ))
= h2k k Lk f kL2 (R+ ,J α,β (x)dx) .
Calculating the supremum with respect to all h ∈ (0, t], we obtain
ωk (f, t)2 ≤ t2k kLk f k2 .
5
Inverse Theorems of Approximation Theory...
Lemma 3.2 For j ≥ 1 we have
j
2
X
22k(j−1) E2j (f )2 ≤
l2k−1 El (f )2 .
l=2j−1 +1
Proof: Note that
j
2
X
l2k−1 ≥ (2j−1 )2k−1 2j−1 = 22k(j−1) .
l=2j−1 +1
Since El (f )2 is monotonically decreasing, we conclude that
j
2
X
22k(j−1) E2j (f )2 ≤
l2k−1 El (f )2 .
l=2j−1 +1
Lemma 3.3 For n ∈ N∗ we have
n
c3 X
(j + 1)2k−1 Ej (f )2 .
n2k j=0
2k En (f )2 ≤
Proof: Note that
n
X
j=0
(j + 1)
2k−1
≥
n
X
(j + 1)
j≥ n
−1
2
2k−1
n 2k−1 n
= 2−2k n2k .
≥
2
2
Since Ej (f )2 is monotonically decreasing, we conclude that
2k En (f )2 ≤
n
c3 X
(j + 1)2k−1 Ej (f )2 .
n2k j=0
Lemma 3.4 If Φν ∈ Iν such that kf − Φν k2 = Eν (f )2 For every ν ∈ N,
then
kLk Φ2j+1 − Lk Φ2j k2 ≤ c2 22k(j+1)+1 E2j (f )2 .
In particular
kLk Φ1 k2 = kLk Φ1 − Lk Φ0 k2 ≤ c2 24k+1 E0 (f )2 .
Proof: By lemma (2.2) and the fact that Eν (f )2 is monotonically decreasing
with respect to ν, we obtain
kLk Φ2j+1 − Lk Φ2j k2 ≤
=
≤
≤
c2 22k(j+1) kΦ2j+1 − Φ2j k2
c2 22k(j+1) k(f − Φ2j ) − (f − Φ2j+1 )k2
c2 22k(j+1) (E2j (f )2 + E2j+1 (f )2 )
c2 22k(j+1)+1 E2j (f )2 ,
6
R. Daher et al.
and
kLk Φ1 − Lk Φ0 k2 ≤ c2 kΦ1 − Φ0 k2 = c2 k(f − Φ1 ) − (f − Φ0 )k2
≤ c2 (E1 (f )2 + E0 (f )2 )
≤ 2c2 E0 (f )2 ≤ c2 24k+1 E0 (f )2 .
The following theorems are analogues of the classical inverse theorems of approximation theory due to Stechkin in the case p = ∞ and A.F.Timan in the
case 1 ≤ p < ∞ (see [4] and [5]).
Theorem 3.5 For every function f ∈ L2(α,β) (R+ ) and n ∈ N∗ we have
n
1
c X
ωk (f, )2 ≤ 2k
(j + 1)2k−1 Ej (f )2 ,
n
n j=0
where c = c(k, α, β) is a positive constant.
Proof: Let 2m ≤ n ≤ 2m+1 for any integer m ≥ 0.
For every ν ∈ N, let Φν ∈ Iν such that kf − Φν k2 = Eν (f )2 . By formulas (i)
and (ii ) of lemma (3.1), we obtain
1
1
1
1
ωk (f, )2 ≤ ωk (f − Φ2m+1 , )2 + ωk (Φ2m+1 , )2 ≤ 2k kf − Φ2m+1 k2 + ωk (Φ2m+1 , )2 .
n
n
n
n
Therefore
1
1
1
ωk (f, )2 ≤ 2k E2m+1 (f )2 + ωk (Φ2m+1 , )2 ≤ 2k En (f )2 + ωk (Φ2m+1 , )2 . (7)
n
n
n
Now with the aid of lemmas (3.2), (3.4) and formula (iii ) of lemma (3.1), we
conclude that
1
1
ωk (Φ2m+1 , )2 ≤
kLk Φ2m+1 k2
n
n2k
!
m
X
1
kLk Φ1 − Lk Φ0 k2 +
kLk Φ2j+1 − Lk Φ2j k2
≤
n2k
j=0
!
m
X
c2
4k+1
2k(j+1)+1
≤
2
E0 (f )2 +
2
E2j (f )2
n2k
j=0
!
m
X
c2 4k+1
≤
2
E0 (f )2 +
22k(j−1) E2j (f )2
n2k
j=0


j
m
2
X X
c2 4k+1 
≤
2
E
(f
)
+
E
(f
)
+
l2k−1 El (f )2 
0
2
1
2
n2k
j=1 l=2j−1 +1
!
2m
X
c2 4k+1
≤
2
E0 (f )2 + E1 (f )2 +
(j + 1)2k−1 Ej (f )2 .
n2k
j=2
7
Inverse Theorems of Approximation Theory...
Whence
m
2
1
c4 X
ωk (Φ2m+1 , )2 ≤
(j + 1)2k−1 Ej (f )2 .
n
n2k j=0
(8)
Thus from (7) and (8) we derive the estimate
n
1
c4 X
k
(j + 1)2k−1 Ej (f )2 .
ωk (f, )2 ≤ 2 En (f )2 + 2k
n
n j=0
(9)
By lemma (3.3) and formula (9), we have
n
1
c X
(j + 1)2k−1 Ej (f )2 .
ωk (f, )2 ≤ 2k
n
n j=0
Theorem 3.6 Suppose that f ∈ L2(α,β) (R+ ) and
∞
X
j 2m−1 Ej (f )2 < ∞.
j=1
m
Then f ∈ W2,(α,β)
and for n ∈ N∗ , we have
1
ωk (Lm f, )2 ≤ C
n
n
∞
X
1 X
2(k+m)−1
(j
+
1)
E
(f
)
+
j 2m−1 Ej (f )2
j
2
n2k j=0
j=n+1
!
.
where C = c(k, m, α, β) is a positive constant.
Proof: Let 2m ≤ n ≤ 2m+1 for any integer m ≥ 0.
By lemmas 3.4 and 3.2, we have for r ≤ m
∞
X
j=0
r
r
kL Φ2j+1 − L Φ2j k2 ≤ c2
∞
X
22r(j+1)+1 E2j (f )2
j=0
2r+1
= c2 2
4r+1
E1 (f )2 + c2 2
∞
X
22r(j−1) E2j (f )2
j=1
≤ c2 24r+1 E1 (f )2 +
∞
X
!
22r(j−1) E2j (f )2
j=1

≤ c2 24r+1 E1 (f )2 +
∞
X
j
2
X
j=1 l=2j−1 +1
≤ c2 24r+1
∞
X
j=1
j 2r−1 Ej (f )2 < ∞.

l2r−1 El (f )2 
8
R. Daher et al.
Note that
f = Φ1 +
∞
X
(Φ2j+1 − Φ2j ).
j=0
Since the series
∞
X
Lr Φ2j+1 − Lr Φ2j converges in L2(α,β) (R+ ) and L is a closed
j=0
operator, we have
L r f = L r Φ1 +
∞
X
(Lr Φ2j+1 − Lr Φ2j ).
j=0
m
Whence Lr f ∈ L2(α,β) (R+ ) for r ≤ m and f ∈ W2,(α,β)
.
By formula (i) of lemma (3.1), we obtain
1
1
1
ωk (Lm f, )2 ≤ ωk (Lm f − Lm Φ2s+1 , )2 + ωk (Lm Φ2s+1 , )2 .
n
n
n
Using lemmas (3.1) , (3.2) and (3.4), we get
1
ωk (Lm f − Lm Φ2s+1 , )2 ≤ 2k kLm f − Lm Φ2s+1 k2
n
∞
X
k
≤ 2
kLm Φ2j+1 − Lm Φ2j k2
j=s+1
∞
X
≤ 2k c2
22m(j+1)+1 E2j (f )2
j=s+1
k+4m+1
≤ c2 2
∞
X
22m(j−1) E2j (f )2
j=s+1
≤ c2 2k+4m+1
∞
X
j
2
X
l2m−1 El (f )2
j=s+1 l=2j−1 +1
∞
X
1
j 2m−1 Ej (f )2 .
ωk (Lm f − Lm Φ2s+1 , )2 ≤ c2 2k+4m+1
n
j=2s +1
Whence
∞
X
1
ωk (Lm f − Lm Φ2s+1 , )2 ≤ c5
j 2m−1 Ej (f )2 .
n
j=2s +1
(10)
9
Inverse Theorems of Approximation Theory...
Now with the aid of lemmas (3.2) , (3.4) and by formula (iii ) of lemma (3.1),
we conclude that
1
1
ωk (Lm Φ2s+1 , )2 ≤
kLm+k Φ2s+1 k2
n
n2k
!
s
X
1
≤
kLm+k Φ1 − Lm+k Φ0 k2 +
kLm+k Φ2j+1 − Lm+k Φ2j k2
n2k
j=0
!
s
X
c2
≤
24(k+m)+1 E0 (f )2 +
22(k+m)(j+1)+1 E2j (f )2
2k
n
j=0
!
s
X
c2 4(k+m)+1
2
E0 (f )2 +
22(k+m)(j−1) E2j (f )2
≤
n2k
j=0


s
2j
X
X
c2 4(k+m)+1 
l2(k+m)−1 El (f )2 
2
E0 (f )2 + E1 (f )2 +
≤
n2k
j=1 l=2j−1 +1
!
2s
X
c2 4(k+m)+1
≤
2
E0 (f )2 + E1 (f )2 +
(j + 1)2(k+m)−1 Ej (f )2 .
n2k
j=2
Whence
s
2
1
c6 X
m
ωk (B Φ2s+1 , )2 ≤
(j + 1)2(k+m)−1 Ej (f )2 .
n
n2k j=0
(11)
Thus from (10) and (11) we derive the estimate
1
ωk (B m f, )2 ≤ C
n
∞
X
n
1 X
2m−1
j
(j + 1)2(k+m)−1 Ej (f )2
Ej (f )2 + 2k
n
j=n+1
j=0
!
.
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10
R. Daher et al.
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