# New Results on the Equivalence ofK-Functionals and Modulus of Continuity of Functions Defined on the Sobolev Space Constructed by the Generalized Jacobi-Dunkl Operator

Authors: Ali El Mfadel; Said Melliani; M’hamed Elomari
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

In this paper, we establish some new generalized results on the equivalence ofK-functionals and modulus of continuity of functions defined on the Sobolev space Lα,β2ℝ, by using the harmonic analysis related to the Jacobi-Dunkl operator Δα,β, where α≥β≥−1/2 and α≠−1/2.


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## Body

## 1. Introduction

The modulus of continuity plays a fundamental role in the approximation theory of functions. For a given positive real numberr and a positive integer m, the classical modulus of continuity of a function f∈L2ℝ is defined by
(1)ωmf,r=sup0<h≤rDhmfL2,where Dhmf=Th−Imf, such that I is the unit operator in L2ℝ and Th represents the usual translation operator given by Thfx=fx+h.The classicalK-functional, introduced by Peetre in [1], is defined by
(2)Kpf,r=inff−gL2+rDpgL2;g∈W2p,where W2p is the Sobolev space constructed by the operator D=d/dx and
(3)W2p≔f∈L2ℝ:Dif∈L2ℝ,i=1,⋯..,p.A remarkable result of the approximation theory of functions onℝ which establishes the equivalence between the modulus of continuity and the K-functional can be formulated as follows:Theorem 1 (see [2]).
There exist two positive real constantsC1 and C2 such that for any function f∈L2ℝ and r>0, we have
(4)C1ωpf,r≤Kpf,rp≤C2ωpf,r.The translation operatorfx↦fx+y is used for the construction of modulus of continuity and smoothness which are the fundamental elements in the classical theory of the approximation of functions on ℝ (see [3–5]).The equivalence between the modulus of continuity and theK-functionals has been established in [6]. Various generalized continuity moduli are studied in [7, 8]. A considerable attention has been devoted to finding generalizations of Theorem 1 (see, for example, [9–12]). This theorem was recently generalized in [13], for the Jacobi transform in the space L2ℝ+,Aα,βxdx by using a generalized Jacobi translation operator. Many generalized moduli of smoothness are often more convenient than the usual ones for the study of the connection between the smoothness properties of a function and the best approximations of this function in weight functional spaces (see, for example, [14]).Motivated by the results mentioned above, in this paper, we establish an analogue result of Theorem1 by using the harmonic analysis associated with the Jacobi-Dunkl operator. The usual translation operators are replaced by generalized Jacobi-Dunkl translation operators in our new result.Our paper is organized as follows: Section2 gives some basic definitions, notations, lemmas, and theorems as preliminaries. Our main result is given in Section 3 followed by conclusion and future works in Section 4.


## 2. Preliminaries

In this section, we give some notations, definitions, and results of the harmonic analysis related to the Jacobi-Dunkl operatorΛα,β defined by
(5)Λα,βfx≔ddxfx+2α+1cothx+2β+1tanhx×fx−f−x2,where α≥β≥−1/2 and α≠−1/2. We cite here, as briefly as possible, only those properties actually required for the discussion; for more details, we refer the reader to [15, 16].The Jacobi-Dunkl Laplacian operator defined onℝ is given by
(6)Δα,βfx≔Λα,β2fx,which is equivalent to
(7)Δα,βfx=Lα,βfx+ddxΛα,β′Λα,βfxfx−f−x2,where Lα,β is the Jacobi operator defined by
(8)Lα,βfx=d2dx2fx+2α+1cothx+2β+1tanhxddxfx.In the rest of the paper, we denote the following:(i)
Dℝ: the space of indefinitely differentiable functions on ℝ with compact support(ii)
Dpℝ: the space of indefinitely differentiable pair functions on ℝ with compact support(iii)
Sℝ: the usual Schwartz space of indefinitely differentiable functions rapidly decreasing on ℝ, as well as their derivatives of all orders, provided with the topology of the following seminorms(9)Lm,nf=supx∈ℝ,0≤k≤n∣1+x2m∣dkdxkfx∣<+∞,S1ℝ=coshx−2ρf;f∈Sℝ.This space is provided with the topology of the following seminormsLm,n1, m,n∈ℕ2, where
(10)Lm,n1f=supx∈ℝ,0≤k≤ncoshx−2ρ1+x2mdkdxkfx<+∞.(iv)
Lα,βpℝ, p≥1, and the space of measurable functions f on ℝ such that(11)fp,α,β=∫ℝfxpAα,βxdx1/p<+∞,if1≤p<+∞,essupx∈ℝfx<+∞,ifp=+∞.where
(12)Aα,βx=22α+β+1sinhx2α+1coshx2β+1.(v)
W2,α,βm is the Sobolev space constructed by the generalized Laplacian Jacobi-Dunkl operator, i.e.,(13)W2,α,βm=f∈Lα,β2ℝ:Δα,βif∈Lα,β2ℝ,i=1,2,⋯,m,where Δα,β0f=f; Δα,βif=Δα,βΔα,βi−1f, i=1,2,⋯,m.Letf∈Lα,β2ℝ and r>0.
(vi)
The generalized modulus of continuity of the functionf at order m is defined by the following formula:(14)ωmf,r2,α,β=sup0<h≤rAhmf2,α,β,where
(15)Ahmf=Th+T−h−2Imf,m=1,2,⋯,and I is the unit operator of the space Lα,β2ℝ.(vii)
Gauss’s hypergeometric functionF21a,b;c;z is defined by(16)F21a,b;c;z≔∑l=0+∞alblcll!zl,z<1,wherea,b,c,z∈ℂ, c∉ℤ−, and al is the Pochhammer symbol given by
(17)al≔aa+1⋯a+l−1,ifl∈ℕ∗,1,ifl=0.where ℕ∗ is the set of nonzero positive integers.
(viii)
The Jacobi functionφμα,β is defined by(18)φμα,βx=2F1ρ+iμ2,ρ−iμ2;α+1;−sinhx2.(ix)
The generalizedK-functional of a function f is defined by(19)Kf,r,Lα,β2ℝ,W2,α,βm=inff−g2,α,β+rΔα,βmg2,α,β:g∈W2,α,βm,whereΔα,β is the generalized Jacobi-Dunkl Laplacian operator defined by (6).For the short form, we denote(20)Kmf,r2,α,β=Kf,r,Lα,β2ℝ,W2,α,βm.(x)
The Jacobi-Dunkl kernelψλα,β∈C∞ℝ is the unique solution of the following differential equation:(21)Λα,βu=iλu,λ∈ℂ,u0=1.which is defined by(22)ψλα,βx≔φμα,βx−iλddxφμα,βx,ifλ∈ℂ∗,1,ifλ=0,whereλ2=μ2+ρ2, ρ=α+β+1, and φμα,β is the Jacobi function given by (18).
(xi)
The Jacobi-Dunkl transform of a functionf∈Dℝ is defined by(23)Fα,βfλ=∫ℝfxψλα,βxAα,βxdx.Property 2 (see [17]).
(1)
Iff∈Lα,β1ℝ, then Fα,βf∈Lσ∞ℝ and we have(24)Fα,βf∞,σ≤f1,α,β.(2)
Iff∈S1ℝ, then(25)Fα,βΛα,βfx=ixFα,βfx,Fα,βΔα,βfx=−x2Fα,βfx.Theorem 3 (see [17]).
(i)
The Jacobi-Dunkl transformFα,β is a topological isomorphism between S1ℝ and Sℝ(ii)
The Jacobi-Dunkl transformFα,β is a topological isomorphism from Lα,β2ℝ to Lσ2ℝ, and we have(26)∫ℝfx2Aα,βxdx=∫ℝFα,βfλ2dσλ.Theorem 4 (see [17]).
Letf∈Lα,β1ℝ such that Fα,βf∈Lσ1ℝ; then, we have
(27)fx=∫ℝFα,βfλψ−λα,βxdσλ.Definition 5.
Letf∈Lα,β2ℝ and ν>0; we define the function Pν by
(28)Pνfx=∫−ν2+ρ2ν2+ρ2Fα,βfλψ−λα,βxdσλ=Fα,β−1Fα,βfλχνλ,whereχνλ is the characteristic function defined by
(29)χνλ=1,ifλ≤ν2+ρ2,0,ifλ>ν2+ρ2,andFα,β−1 is the inverse Jacobi-Dunkl transform.Remark 6.
We can easily prove that the functionPνf is indefinitely differentiable and Pνf∈W2,α,βm, ∀m∈ℕ.Lemma 7 (see [18]).
Letf∈W2,α,βm and r>0; then,
(1)
Fα,βΔα,βmfξ=−1mξ2mFα,βfξ, for all ξ∈ℝ(2)
ωmf,r2,α,β≤2mr2m∥Δα,βmf∥2,α,βLemma 8 (see [18]).
Letf∈Lα,β2ℝ and h>0; then,
(30)Ahmf2,α,β≤10mf2,α,β.


## 3. Main Result

In this section, we give the main result of our paper which is a generalization of Theorem1.Theorem 9.
There exist two positive real constantsC1 and C2 such that for any function f∈Lα,β2ℝ and positive constant r>0, we have
(31)c1ωmf,r2,α,β≤Kmf,r2m2,α,β≤c2ωmf,r2,α,β.
In order to prove Theorem9, we will need some preliminary results.Proposition 10.
Iff∈W2,α,βm and h>0, then
(32)Fα,βAhmfλ=2mφμα,βh−1mFα,βfλ.Proof.
By using (18) and (23), we obtain
(33)Fα,βAhfλ=Fα,βThf+T−hf−2fλ,=ψλα,βh+ψλα,β−h−2Fα,βfλ,=2φμα,βh−1Fα,βfλ.
We use the proof by induction form, and we obtain the result.Proposition 11.
Letf∈W2,α,βm and r>0; then, we have
(34)ωmf,r2,α,β≤2mr2mΔα,βmf2,α,β.Proof.
Suppose thath∈0,r. By using Lemmas 7 and 8, we have
(35)Ahmf2,α,β2=Fα,βAhmf2,σ2,=∫ℝFα,βAhmfλ2dσλ,=22m∫ℝφμα,βh−12mFα,βfλ2dσλ,≤22mh4m∫ℝλ4mFα,βfλ2dσλ,=22mh4m∫ℝ−1mλ2mFα,βfλ2dσλ,=22mh4m∫ℝFα,βΔα,βmfλ2dσλ,=22mh4mFα,βΔα,βmf2,σ2,=22mh4mΔα,βmf2,α,β2,thus,
(36)Ahmf2,α,β≤2mh2mΔα,βmf2,α,β,≤2mr2mΛα,βmf2,α,β.
By taking the supremum onh∈0,r, we obtain the result.☐Proposition 12.
Letf∈Lα,β2ℝ and ν>0; then, there exists a positive number c3 such that
(37)f−Pνf2,α,β≤c3A1/νmf2,α,β.Proof.
We use the Plancherel formula (26); we have
(38)f−Pνf2,α,β=Fα,βf−Pνf2,σ=1−χνλFα,βfλ2,σ=1−χνλ1−φμα,β1/νm1−φμα,β1νmFα,βfλ2,σ.
We useλ2=μ2+ρ2; then, we have
(39)λ>ν2+ρ2⇔λ2>ν2+ρ2,⇔μ2>ν2,⇔μ>ν.
There existsc>0 such that
(40)1−φμα,β1ν≥c,for μ∈ℝ such that μ≥ν, and by using Lemma 8, we have
(41)f−Pνf2,α,β≤c−m1−φμα,β1vmFα,βfλ2,σ=c−m2−mFα,βA1/νmfλ2,σ=c3A1/νmf2,α,β.Proposition 13.
Letf∈Lα,β2ℝ and ν>0.
There exists a positive numberc4 such that
(42)Δα,βmPνf2,α,β≤c4ν2mA1/νmf2,α,β.Proof.
We use the Plancherel formula (26) and Lemma 7; we have
(43)Δα,βmPνf2,α,β=Fα,βΔα,βmPνf2,σ,=λ2mχνλFα,βfλ2,σ,=∥λ2m1−φμα,β1/vm1−φμα,β1vmχνλFα,βfλ2,σ.
Forλ>ρ, we obtain
(44)supλ∈ℝ,λ>ρλ2mχνλ1−φμα,β1/vm=ν2msupλ∈ℝ,λ>ρλ/ν2mχνλ1−φμα,β1/vm,=ν2msupλ≤ν2+ρ2,λ>ρλ/ν2m1−φμα,β1/vm.
Note that ifν is large enough, then
(45)λ/ν2m1−φμα,β1/vm⟶4mα+1m,thus
(46)c4=supρ<∣λ∣≤ν2+ρ2λ/ν2m1−φμα,β1/νm.
Finally, formula (42) is proven.Remark 14.
Letf∈Lα,β2ℝ and ν>0. There exists a positive constant c4 such that
(47)Δα,βmPνf2,α,β≤c4ν2mωmf,1ν2,α,β.Proof of Theorem 9.
(i)
We start by giving the proof of the inequality:(48)c1ωmf,r2,α,β≤Kmf,r2m2,α,β.
For this purpose, leth∈0,r and g∈W2,α,βm.
By using Propositions10 and 11, we have
(49)Ahmf2,α,β=Ahmf−g+g2,α,β,≤Ahmf−g2,α,β+Ahmg2,α,β,≤10mf−g2,α,β+2mh2mΔα,βmg2,α,β,≤10mf−g2,α,β+2mr2mΔα,βmg2,α,β,≤10mf−g2,α,β+r2mΔα,βmg2,α,β.
We calculate the supremum onh∈0,r and the infimum on g∈W2,α,βm; we deduce that
(50)c1ωmf,r2,α,β≤Kmf,r2m2,α,β,where c1=10−m.
(ii)
Now, we give the proof of the inequality:(51)Kmf,r2m2,α,β≤c2ωmf,r2,α,β.
SincePνf∈W2,α,βm, then by using the definition of K-functional, we obtain
(52)Kmf,r2m2,α,β≤f−Pνf2,α,β+r2mΔα,βmPνf2,α,β.
We use Remark14; we have
(53)Kmf,r2m2,α,β≤c3ωmf,1ν2,α,β+c4rν2mωmf,1ν2,α,β.
Sinceν is a positive arbitrary value, then by setting ν=1/r, we obtain
(54)Kmf,r2m2,α,β≤c2ωmf,r2,α,β,where c2=c3+c4.


## 4. Conclusion

In this paper, we generalized and we proved some new results on the equivalence between theK-functionals and the modulus of continuity of functions defined on the Sobolev space by using the harmonic analysis related to the Jacobi-Dunkl operator.



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