On the approximation by entire functions of exponential type in Lp,α(R)

DOI 10.1007/s11868-016-0160-1

On the approximation by entire functions of exponential type in L _{p ,α} ( R )

R. Daher ¹ · S. El Ouadih ¹

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 > ⁻ ₂ ¹ , the space of functions f defined on R endowed with the following finite norm

B S. El Ouadih

salahwadih@gmail.com

1

Department of Mathematics, Faculty of Sciences Aïn Chock, University Hassan II, Casablanca,

Morocco

f p ,α :=

R | f ( x )| ^p d μ α ( x )

, 1 ≤ p < ∞,

where the measure d μ α is defined by d μ α (x) = (2 ^{α+ 1} (α + 1)) ^{− 1} |x| ^{2 α+ 1} d x, and L _∞,α denote the space of essentially bounded functions with the finite norm

f _∞,α := ess sup

x ∈R | f (x)|.

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 D _α defined as

D _α f (x) = d f

d x (x) + (α + 1

2 ) f ( x ) − f (− x )

x , f ∈ C ¹ (R).

Since for all ϕ, ψ ∈ S, D _α ϕ, ψ = −ϕ, D _α ψ, where f, g :=

R f (x)g(x)dμ _α (x),

then, D _α u of the distribution u may be defined by the formula D α u, ϕ = −u , D _α ϕ, u ∈ S , ϕ ∈ S . It is obvious that the space L p ,α (R) is embedded into S .

Hence D _α is defined on f ∈ L p ,α (R) . Generally speaking, D _α f is a distribution.

Let j _α ( x ) denote the normalized Bessel function of the first kind of order α given by

j _α ( x ) = (α + 1 ) ∞ n = 0

(−1) ⁿ n!(n + α + 1) ( x

2 ) ²ⁿ . We understand a generalized exponential function as the function

E _α (x) = j _α (i x) + x

2(α + 1) j _α+ 1 (i x), where i = √

− 1. The function y = E _α ( x ) satisfies the equation D _α y = y with the initial data y ( 0 ) = 1 and it is the unique solution (see [5]). Using the correlation

j _α (x) = − x j _α+ 1 (x) 2(α + 1) ,

we conclude that the function E _α (x) admits the representation E _α (x) = j _α (i x) + i j _α (i x).

The Dunkl transform of order α for f ∈ L 1 ,α (R) is defined by F _α f (x) =

R f (y)E _α (−i x y)dμ _α (y), x ∈ R,

and for all f ∈ L 1 ,α (R) such that F _α f ∈ L 1 ,α (R) the inverse Dunkl transform is defined by (see [6])

f ( y ) =

R F α f ( x ) E _α ( i x y ) d μ α ( x ).

The distributional Dunkl transform F α u for a tempered distribution u ∈ S is defined by

F _α u, ϕ = u, F _α ϕ, ϕ ∈ S. (1)

Given s ∈ R, the generalized translation operator T ^s on L 2 ,α (R) is defined by F _α (T ^s f )(x) = E _α (−i sx )F _α 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), F _α 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 trans- form:

F _α (T ^s u )(x) = E _α (−i sx )F _α u(x), T ^s (F _α u)(x) = F _α (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 ^k _h f of order k ( k ∈ N) with the step h > 0 as follows:

^k _h f (x) = (I − T ^h ) ^k f (x) = k

l = 0

(−1) ^l ( ^k _l )T ^lh f (x). (6)

where I is unit operator.

For 1 ≤ p, q < ∞ such that ¹ _p + ¹ _q = 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)dμ _α (y), x ∈ R. (7)

Lemma 2.1 (see [4]) Let p, q , r ≥ 1 and ¹ _r + 1 = ¹ _p + _q ¹ . Assume that f ∈ L p ,α (R), g ∈ L q ,α (R). Then f ∗ α g ∈ L r ,α (R), and

f ∗ _α g r ,α ≤ 3 f p ,α g q ,α .

For σ > 0, denote by M ^σ p ,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 M ^σ p ,α

make a natural approximation tool in L p ,α (R). The best approximation in M ^σ p ,α for f ∈ L p ,α (R) is defined as below:

E _σ ( f ) p ,α := inf{ f − φ p ,α : φ ∈ M ^σ p ,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 M ^σ _{p ,α} if and only if f ∈ L p ,α (R) and F _α f is supported in [−σ, σ ] (i.e, F _α f, ϕ = 0, for all ϕ ∈ S with suppϕ ⊆ [−σ, σ ] ^c ).

Proof (see [[3], Theorem 12 and Remark 13(1)]).

Let W _p ^r _,α be the Sobolev space constructed by the differential-difference operator D _α as follows:

W ^r _{p ,α} = { f ∈ L p ,α (R) : D _α ⁱ f ∈ L p ,α (R), i = 1, . . . , r},

where D _α ⁱ f = D _α (D ⁱ _α ^{− 1} f ) and D ⁰ _α f = f .

Lemma 2.3 If f ∈ W _p ^r _,α (1 ≤ p < ∞), and k ≥ r, k, r ∈ N, then ^k _h f p ,α ≤ ch ^r D ^r _α 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 suppF _α g ⊆ [−σ, σ], then f ∗ _α g belongs to M ^σ _{p ,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 suppF _α ( f ∗ _α g) ⊆ [−σ, σ ].

Let ϕ ∈ S and suppϕ ⊆ [−σ, σ] ^c . From formulas (1), (4) and (7), we have F α ( f ∗ α g ), ϕ = f ∗ α g , F α ϕ

=

R ( f ∗ _α g)(x)F _α ϕ(x)dμ _α (x)

=

R

R f (y)T ^x g(−y)dμ _α (y)

F _α ϕ(x)d μ _α (x)

=

R

R T ^y g (−x)F α ϕ(x)dμ α (x)

f (y)d μ α (y)

=

R

R g (− x ) T ^{− y} F α ϕ( x ) d μ α ( x )

f ( y ) d μ α ( y ).

From (5) we obtain F α ( f ∗ α g), ϕ =

R

R g(−x)F α (E α (−i x y)ϕ(x))d μ α (x)

f (y)dμ α (y).

Put φ(−x) = E _α (−i x y)ϕ(x) ∈ S, from (1) we have F _α ( f ∗ _α g), ϕ =

R

R F _α g(x)φ(x)dμ _α (x)

f (y)d μ _α (y).

Since supp φ ⊆ [−σ, σ ] ^c and supp F _α g ⊆ [−σ, σ ], we have

R F _α g(x)φ(x)d μ _α (x) = F _α g, φ = 0.

This prove that F _α ( f ∗ _α g), ϕ = 0, whence supp F _α ( f ∗ _α g) ⊆ [−σ, σ]. By Lemma

2.2, we have f ∗ _α g ∈ M ^σ p ,α .

Theorem 3.2 Let 1 ≤ p < ∞ and f ∈ W ^r _{p ,α} . Then, we have E _σ ( f ) p ,α ≤ c 1

σ ^r D ^r _α f p ,α , where c 1 = c(α, r) is a constant.

Proof We use the scheme of the classical method of approximation by entire functions of exponential type on R ⁿ 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 g ( t ) d μ α t = 1 , (8)

that is, g ∈ M ^σ _{1 ,α} .

Let f ∈ L p ,α (R). Consider the function

σ ( x ) :=

R g ( t )((− 1 ) ^h

σ

f ( x ) + f ( x )) d μ α ( t ).

By using (6), we have

σ (x) :=

R g(t ) _k

l = 0

(−1) ^{l − 1} ( ^k _l )T

f (x) + f (x)

d μ _α (t )

=

R g(t ) _k

l = 1

(−1) ^{l − 1} ( ^k _l )T

f (x)

dμ _α (t)

=

R g(t ) _k

l = 1

d l T

f (x)

d μ α (t ),

where d l = (−1) ^{l − 1} ( ^k _l ) and so _k

l = 1 d l = 1. From (3) and (4) we have

σ (x) = k

l = 1

d l

R g(t )(T ^{− x} f )

− lt σ

d μ _α (t)

= k

l = 1

d l

R g(− σ t

l )T ^{− x} f (t ) σ l

2 α+ 2

d μ α (t )

= k

l = 1

d l σ l

2 α+ 2

R g(− σ t

l )T ^{− x} f (t)dμ _α (t)

= k

l = 1

d l σ l

2 α+ 2

R T ^x g (− σ t

l ) f ( t ) d μ α ( t )

=

R f (t )T ^x K _σ (−t )dμ _α (t) where

K _σ (t ) = k

l = 1

d l σ l

2 α+ 2

g( σt

l ). (9)

From (7) we obtain

σ ( x ) = f ∗ α K _σ ( x ). (10)

Since g ∈ M ^σ _{1 ,α} , then in view of (9) the function K _σ belongs to M ^σ _{1 ,α} . It follows

from (10) and Lemma 3.1 that _σ ∈ M ^σ p ,α .

Let f ∈ W _p ^r _,α and let k = 2r. Using Lemma 2.3 and (8), we obtain that f (x) − _σ (x) p ,α =

f (x) −

R g(t )((−1) ^h

σ

f (x) + f (x))dμ _α (t) p ,α

= f (x) +

R g(t ) ^h

σ

f (x)dμ _α (t ) − f (x)

R g(t)dμ _α (t) p ,α

=

R g ( t ) ^h

σ

f ( x ) d μ α ( t ) p ,α

≤

R g(t ) ^h

σ

f p ,α dμ _α (t)

≤ c

σ ^r D _α ^r f p ,α

R g (t )t ^r dμ α (t).

If we choose the function g such that

R g(t)t ^r d μ _α (t) is finite, then E _σ ( f ) p ,α ≤ c 1

σ ^r D ^r _α f p ,α , where c 1 = c

R g ( t ) t ^r d μ α ( t ) . We can take

g (t ) = γ sin _N ^t

t N

,

where N ∈ N ^∗ such that N ≥ r + 2α + 3 and γ is a constant, for which (8) holds.

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