J. Pseudo-Differ. Oper. Appl. DOI 10.1007/s11868-016-0160-1
On the approximation by entire functions of exponential type in L p,α (R) R. Daher1 · S. El Ouadih1
Received: 20 November 2015 / Accepted: 1 May 2016 © Springer International Publishing 2016
Abstract In this paper, we obtain the direct theorem of approximation of functions on the real line R in the metric of L p with some power weight using generalized Dunkl shifts. Keywords Dunkl transform · Generalized translation · Generalized convolution · Entire functions of exponential type
1 Introduction For the real-line R, the generalized Dunkl shift operator is one of the most important, and is used in the study of various problems involving Dunkl differential operators. Dunkl harmonic analysis, which deals with Dunkl integral transformations and their applications, is closely connected with the generalized Dunkl shift. In the present paper, we prove that functions on the real line in a Sobolev-type space can be approximated by entire functions of exponential type in the weighted L p -space.
2 The Dunkl transform and its basic properties Let L p,α (R), 1 ≤ p ≤ ∞ and α > with the following finite norm
B 1
−1 2 , the space of functions
f defined on R endowed
S. El Ouadih
[email protected] Department of Mathematics, Faculty of Sciences Aïn Chock, University Hassan II, Casablanca, Morocco
R. Daher, S. El Ouadih
f p,α :=
1
R
| f (x)| p dμα (x)
p
, 1 ≤ p < ∞,
where the measure dμα is defined by dμα (x) = (2α+1 (α + 1))−1 |x|2α+1 d x, and L ∞,α denote the space of essentially bounded functions with the finite norm f ∞,α := ess sup | f (x)|. x∈R
We denote by C k (R) the set of k times continuously differentiable functions on R. S is the Schwartz function space defined on R with S as its dual space. The Dunkl operator is differential-difference operator Dα defined as Dα f (x) =
1 f (x) − f (−x) df (x) + (α + ) , dx 2 x
f ∈ C 1 (R).
Since for all ϕ, ψ ∈ S, Dα ϕ, ψ = −ϕ, Dα ψ, where f, g := f (x)g(x)dμα (x), R
then, Dα u of the distribution u may be defined by the formula Dα u, ϕ = −u, Dα ϕ, u ∈ S , ϕ ∈ S. It is obvious that the space L p,α (R) is embedded into S . Hence Dα is defined on f ∈ L p,α (R). Generally speaking, Dα f is a distribution. Let jα (x) denote the normalized Bessel function of the first kind of order α given by jα (x) = (α + 1)
∞ n=0
x (−1)n ( )2n . n!(n + α + 1) 2
We understand a generalized exponential function as the function E α (x) = jα (i x) +
x jα+1 (i x), 2(α + 1)
√ where i = −1. The function y = E α (x) satisfies the equation Dα y = y with the initial data y(0) = 1 and it is the unique solution (see [5]). Using the correlation jα (x) = −
x jα+1 (x) , 2(α + 1)
we conclude that the function E α (x) admits the representation E α (x) = jα (i x) + i jα (i x). The Dunkl transform of order α for f ∈ L 1,α (R) is defined by Fα f (x) = f (y)E α (−i x y)dμα (y), x ∈ R, R
On the approximation by entire...
and for all f ∈ L 1,α (R) such that Fα f ∈ L 1,α (R) the inverse Dunkl transform is defined by (see [6]) f (y) =
R
Fα f (x)E α (i x y)dμα (x).
The distributional Dunkl transform Fα u for a tempered distribution u ∈ S is defined by Fα u, ϕ = u, Fα ϕ, ϕ ∈ S.
(1)
Given s ∈ R, the generalized translation operator T s on L 2,α (R) is defined by Fα (T s f )(x) = E α (−isx)Fα f (x).
(2)
The linear operator T s can be extended to a continuous operator on L p,α (R) with T s f p,α ≤ 3 f p,α , 1 ≤ p ≤ ∞ (see [7]). If f ∈ L 1,α (R), Fα f ∈ L 1,α (R), then T s f (x) = T −x f (−s).
(3)
On the other hand, since T s ϕ, ψ = ϕ, T −s ψ, for ϕ, ψ ∈ S, we extend the operator T s to distributions by T s u, ϕ := u, T −s ϕ, u ∈ S , ϕ ∈ S.
(4)
The following relations connect the Dunkl generalized translation and the Dunkl transform: Fα (T s u)(x) = E α (−isx)Fα u(x), T s (Fα u)(x) = Fα (E α (i.s)u)(x).
(5)
for any x ∈ R, and any u ∈ S . Given f ∈ L p,α (R), we define the differences kh f of order k(k ∈ N) with the step h > 0 as follows: kh f (x) = (I − T h )k f (x) =
k
(−1)l (lk )T lh f (x).
(6)
l=0
where I is unit operator. For 1 ≤ p, q < ∞ such that 1p + q1 = 1, the generalized convolution of f ∈ L p,α (R) and g ∈ L q,α (R) is defined by f ∗α g(x) :=
R
f (y)T x g(−y)dμα (y), x ∈ R.
(7)
R. Daher, S. El Ouadih
Lemma 2.1 (see [4]) Let p, q, r ≥ 1 and r1 + 1 = g ∈ L q,α (R). Then f ∗α g ∈ L r,α (R), and
1 p
+ q1 . Assume that f ∈ L p,α (R),
f ∗α gr,α ≤ 3 f p,α gq,α . For σ > 0, denote by Mσp,α the space of entire functions of exponential type ≤ σ whose restrictions to R belong to L p,α (R). Then the functions in the space Mσp,α make a natural approximation tool in L p,α (R). The best approximation in Mσp,α for f ∈ L p,α (R) is defined as below: E σ ( f ) p,α := inf{ f − φ p,α : φ ∈ Mσp,α }.
For every u ∈ S , we denote by supp u the support of u, then suppu ⊆ [−σ, σ ] if and only if u, ϕ = 0 for all ϕ ∈ S such that suppϕ ⊆ [−σ, σ ]c . Lemma 2.2 Let σ > 0, 1 ≤ p ≤ ∞ and f ∈ C(R). Then f can be extended to an entire function in Mσp,α if and only if f ∈ L p,α (R) and Fα f is supported in [−σ, σ ] (i.e, Fα f, ϕ = 0, for all ϕ ∈ S with suppϕ ⊆ [−σ, σ ]c ). Proof (see [[3], Theorem 12 and Remark 13(1)]). r be the Sobolev space constructed by the differential-difference operator Let W p,α Dα as follows: r = { f ∈ L p,α (R) : Dαi f ∈ L p,α (R), i = 1, . . . , r }, W p,α
where Dαi f = Dα (Dαi−1 f ) and Dα0 f = f .
r (1 ≤ p < ∞), and k ≥ r, k, r ∈ N, then Lemma 2.3 If f ∈ W p,α
kh f p,α ≤ ch r Dαr f p,α , where c = c(k, r, α) is a constant. Proof (see [1], Lemma 4.2)
3 Main results Lemma 3.1 Let f ∈ L p,α (R) and g ∈ L 1,α (R), 1 ≤ p ≤ ∞. If suppFα g ⊆ [−σ, σ ], then f ∗α g belongs to Mσp,α . Proof Since f ∈ L p,α (R) and g ∈ L 1,α (R), in view of Lemma 2.1 we have f ∗α g ∈ L p,α (R) . We claim that suppFα ( f ∗α g) ⊆ [−σ, σ ].
On the approximation by entire...
Let ϕ ∈ S and suppϕ ⊆ [−σ, σ ]c . From formulas (1), (4) and (7), we have Fα ( f ∗α g), ϕ = f ∗α g, Fα ϕ = ( f ∗α g)(x)Fα ϕ(x)dμα (x) R f (y)T x g(−y)dμα (y) Fα ϕ(x)dμα (x) = R R T y g(−x)Fα ϕ(x)dμα (x) f (y)dμα (y) = R R −y g(−x)T Fα ϕ(x)dμα (x) f (y)dμα (y). = R
R
From (5) we obtain Fα ( f ∗α g), ϕ =
R
R
g(−x)Fα (E α (−i x y)ϕ(x))dμα (x)
f (y)dμα (y).
Put φ(−x) = E α (−i x y)ϕ(x) ∈ S, from (1) we have Fα ( f ∗α g), ϕ =
R
R
Fα g(x)φ(x)dμα (x)
f (y)dμα (y).
Since supp φ ⊆ [−σ, σ ]c and supp Fα g ⊆ [−σ, σ ], we have R
Fα g(x)φ(x)dμα (x) = Fα g, φ = 0.
This prove that Fα ( f ∗α g), ϕ = 0, whence supp Fα ( f ∗α g) ⊆ [−σ, σ ]. By Lemma
2.2, we have f ∗α g ∈ Mσp,α . r . Then, we have Theorem 3.2 Let 1 ≤ p < ∞ and f ∈ W p,α
E σ ( f ) p,α ≤
c1 Dαr f p,α , σr
where c1 = c(α, r ) is a constant. Proof We use the scheme of the classical method of approximation by entire functions of exponential type on Rn that goes back to Bernstein and Nikol’skii (see [2], Ch.5). Let g be a non-negative entire function on R of exponential type 1 such that R
that is, g ∈ Mσ1,α .
g(t)dμα t = 1,
(8)
R. Daher, S. El Ouadih
Let f ∈ L p,α (R). Consider the function σ (x) :=
g(t)((−1)ht f (x) + f (x))dμα (t). σ
R
By using (6), we have σ (x) :=
R
=
R
k lt g(t) (−1)l−1 (lk )T σ f (x) + f (x) dμα (t) l=0
k lt l−1 k σ g(t) (−1) (l )T f (x) dμα (t) l=1
=
R
g(t)
k
k
k
k
dl
l=1
=
k l=1
=
R
k
l=1 dl
= 1. From (3) and (4) we have
lt dμα (t) g(t)(T −x f ) − σ R
dl
l=1
=
f (x) dμα (t),
dl
l=1
=
dl T
l=1
where dl = (−1)l−1 (lk ) and so σ (x) =
lt σ
dl
R
g(−
σ 2α+2 σ t −x )T f (t) dμα (t) l l
σ 2α+2 l
R
σ 2α+2 l
R
g(−
σ t −x )T f (t)dμα (t) l
T x g(−
σt ) f (t)dμα (t) l
f (t)T x K σ (−t)dμα (t)
where K σ (t) =
k l=1
dl
σ 2α+2 l
g(
σt ). l
(9)
From (7) we obtain σ (x) = f ∗α K σ (x).
(10)
Since g ∈ Mσ1,α , then in view of (9) the function K σ belongs to Mσ1,α . It follows from (10) and Lemma 3.1 that σ ∈ Mσp,α .
On the approximation by entire... r and let k = 2r . Using Lemma 2.3 and (8), we obtain that Let f ∈ W p,α
f (x) − σ (x) p,α
h = f (x) − g(t)((−1) t f (x) + f (x))dμα (t) σ R p,α h f (x) + = g(t) f (x)dμ (t) − f (x) g(t)dμ (t) t α α σ
R
R
h = g(t) t f (x)dμα (t) σ R p,α ≤ g(t)ht f p,α dμα (t) σ R c r ≤ r Dα f p,α g(t)t r dμα (t). σ R If we choose the function g such that g(t)t r dμα (t) is finite, then
p,α
R
E σ ( f ) p,α ≤ where c1 = c We can take
R
c1 Dαr f p,α , σr
g(t)t r dμα (t). g(t) = γ
sin t
t N
N ,
where N ∈ N∗ such that N ≥ r + 2α + 3 and γ is a constant, for which (8) holds.
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