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We introduce in this paper the concept of Dunkl-coherent pair of forms (linear functionals) in the symmetric case. We prove that if { u , v } is a Dunkl-symmetrically coherent pair of symmetric for...

In this work, new characterizations of Herz and Herz-type Hardy spaces associated with the Dunkl operator on the real line are introduced.

In a similar way that the Appell sequences of polynomials can be extended to the Dunkl context, where the ordinary derivative is replaced by Dunkl operator on the real line, and the exponential function is replaced by the so-called Dunkl kernel, one can expect that the discrete Appell sequences can be extended to the Dunkl context. In this extension, the role of the ordinary translation is played by the Dunkl translation, that is a much more intricate operator. In this paper, we define discrete Appell-Dunkl sequences of polynomials, and we give some properties and examples. In particular, we show which is the suitable definition for the Bernoulli polynomials of the second kind in the Dunkl context.

We consider the Dunkl differential-difference operator associated with a reflection group W and a nonnegative parameter function k. We exhibit and study a new family of sharp integral inequalities ...

For $$p\ge p_0:=2\lambda /(2\lambda +1)$$ with $$\lambda >0$$ , the Hardy space $$H_\lambda ^p({\mathbb {R}}^2_+)$$ associated with the Dunkl transform $${\mathscr {F}}_{\lambda }$$ and the Dunkl operator D on the line $${\mathbb {R}}$$ , where $$(D_xf)(x)=f'(x)+\frac{\lambda }{x} (f(x)-f(-x))$$ , is the set of functions $$F=u+iv$$ on the half plane $${\mathbb {R}}^2_+=\{(x,y):\,x\in {\mathbb {R}}, y>0\}$$ , satisfying the generalized Cauchy–Riemann equations $$D_xu-\partial _yv=0$$ , $$\partial _yu+D_xv=0$$ , and $$\sup _{y>0}\int _{{\mathbb {R}}}| F(x,y)|^p|x|^{2\lambda }dx

In the paper we study the Schrodinger oscillatory integrals $$T^t_{\lambda ,a}f(x)$$ ( $$\lambda \ge 0$$ , $$a>1$$ ) associated with the one-dimensional Dunkl transform $${\mathscr {F}}_{\lambda }$$ . If $$a=2$$ , the function $$u(x,t):=T^t_{\lambda ,2}f(x)$$ solves the free Schrodinger equation associated to the Dunkl operator, with f as the initial data. It is proved that, if f is in the Sobolev spaces $$H^s_{\lambda }({\mathbb {R}})$$ associated with the Dunkl transform, with the exponents s not less than 1 / 4, then $$T^t_{\lambda ,a}f$$ converges almost everywhere to f as $$t\rightarrow 0$$ . A counterexample is constructed to show that 1 / 4 can not be improved for $$a=2$$ , and when $$1/4\le s\le 1/2$$ , the Hausdorff dimension of the divergence set of $$T^t_{\lambda ,a}f$$ for $$f\in H_{\lambda }^s({\mathbb {R}})$$ is proved to be $$1-2s$$ at most.

Using the results in [12] where a construction of the Dunkl intertwining operator for a large set of regular parameter functions is provided, we establish an integral expression for the Dunkl kernel in the context of the dihedral group 𝒟 n {\mathcal{D}_{n}} with constant parameter function k ∈ ℂ {k\in\mathbb{C}} and arbitrary order n ≥ 2 {n\geq 2} . Our main tool is a differential system that leads to the explicit expression of the Dunkl kernel whenever an appropriate solution of it is obtained. In particular, an explicit expression of the Dunkl kernel E k ⁢ ( x , y ) {E_{k}(x,y)} is given when one of its argument x or y is invariant under the action of any reflection in the dihedral group. We obtain also a generating series for the homogeneous components K m ⁢ ( x , y ) {K_{m}(x,y)} , m ∈ ℤ + {m\in\mathbb{Z}^{+}} , of the Dunkl kernel and provide new sharp estimates for the Dunkl kernel in the large context k ∈ ℂ {k\in\mathbb{C}} , n ≥ 2 {n\geq 2} and - 2 ⁢ n ⁢ k ≠ 1 , 2 , 3 , … {-2nk\neq 1,2,3,\dots\,} .

In the present paper, a new family of integral transforms depending on two parameters and related to the Dunkl transform is introduced. Well-known transforms, such as the fractional Dunkl transform, Dunkl transform, linear canonical transform, canonical Hankel transform, Fresnel transform, etc., can be seen to be special cases of this general transform. Some useful properties of the considered transform such as Riemann–Lebesgue lemma, reversibility property, additivity property, operational formula, Plancherel formula, Bochner type identity and master formula are derived. The intimate connection that exists between this transformation and the quantum harmonic oscillator is developed.

In this paper the notion of a Dunkl two-wavelet is introduced. The resolution of the identity formula for the Dunkl continuous wavelet transform is then formulated and proved. Calderon’s type reproducing formula in the context of the Dunkl two-wavelet theory is proved. The two-wavelet localization operators in the setting of the Dunkl theory are then defined. The Schatten–von Neumann properties of these localization operators are established, and for trace class localization operators, the traces and the trace class norm inequalities are presented. It is proved that under suitable conditions on the symbols and two Dunkl wavelets, the boundedness and compactness of these localization operators on \(L^{p}_{k}(\mathbb {R}^{d})\), \(1 \le p \le \infty \). Finally typical examples of localization operators are presented.

In this paper we show how to use mollifiers to regularize functions relative to a set of Dunkl operators in R d with Coxeter–Weyl group W, multiplicity function k and weight function ω k . In particular for Ω a W-invariant open subset of R d , for ϕ ∈ D ( R d ) a radial function and u ∈ L loc 1 ( Ω , ω k ( x ) d x ) , we study the Dunkl-convolution product u ⁎ k ϕ and the action of the Dunkl-Laplacian and the volume mean operators on these functions. The results are then applied to obtain an analog of the Weyl lemma for Dunkl-harmonic functions and to characterize them by invariance properties relative to mean value and convolution operators.

The purpose of this paper is to introduce a modification of q-Dunkl generalization of exponential functions. These types of operators enable better error estimation on the interval $[\frac{1}{2},\infty)$ than the classical ones. We obtain some approximation results via a well-known Korovkin-type theorem and a weighted Korovkin-type theorem. Further, we obtain the rate of convergence of the operators for functions belonging to the Lipschitz class.

In this paper, we establish new spectral theorems for the Dunkl transform on \(\mathbb {R}^{d}\).

In this paper we introduce q-Szász-Mirakjan-Kantorovich operators generated by a Dunkl generalization of the exponential function and we propose two different modifications of the q-Szász-Mirakjan-Kantorovich operators which preserve some test functions. We obtain some approximation results with the help of the well-known Korovkin theorem and the weighted Korovkin theorem for these operators. Furthermore, we study convergence properties in terms of the modulus of continuity and the class of Lipschitz functions. This type of operator modification enables better error estimation than the classical ones. We also obtain a Voronovskaja-type theorem for these operators.

The aim of this paper is to prove Heisenberg-Pauli-Weyl inequality for a fractional power of the Dunkl transform on the real line for which there is an index law and a Plancherel theorem.

We consider settings of the Dunkl Laplacian, the Dunkl harmonic oscillator, and the Dunkl Ornstein-Uhlenbeck operator with the underlying group of reflections isomorphic to Zd 2. In each of these contexts, we admit all real-valued multiplicity functions, not necessarily bounded from below, and construct the corresponding ‘exotic’ transform or orthogonal system. This leads to new Dunkl operator-based frameworks, which generalize those yet well known, and in which harmonic analysis can reasonably be developed. To support the last claim, in all the cases we study the associated heat semigroup maximal operators and prove that they are bounded on Lp, p>1, and from L1to weak L2.

We introduce BernoulliDunkl and ApostolEulerDunkl polynomials as generalizations of Bernoulli and ApostolEuler polynomials, where the role of the derivative is now played by the Dunkl operator on the real line. We use them to find the sum of many different series involving the zeros of Bessel functions. © 2018 Elsevier Inc.

We generalize classical Hobson's formula concerning partial derivatives of radial functions on a Euclidean space to a formula in the Dunkl analysis. As applications we give new simple proofs of known results involving Maxwell's representation of harmonic polynomials, Bochner-Hecke identity, Pizzetti formula for spherical mean, and Rodrigues formula for Hermite polynomials.

Using a generalized dual translation operator, we obtain an analog of Titchmarsh’s theorem for the generalized Dunkl transform for functions satisfying the Q-Dunkl Lipschitz condition in the space \(\mathrm {L}_{Q}^{2}\).

We prove an analogue of Beurling's theorem in the setting of Fourier–Dunkl transform, which improves the theorem of Kawazoe–Mejjaoli [Uncertainty principles for the Dunkl transform. Hiroshima Math J. 2010;40(2):241–268].

In Dunkl theory on Rd which generalizes classical Fourier analysis, we study first the behavior at infinity of the Riesz potential of a non compactly supported function. Second, we give for 1, Comment: 22 pages. arXiv admin note: text overlap with arXiv:1301.6267, arXiv:1305.4394, arXiv:1208.5034

In this paper, we introduce new function spaces that are denoted by Dαp, 1≤p≤∞, α≥−12, and that are spaces of DLp type where the Dunkl convolution and the Dunkl transformation are defined. Some properties of these spaces are studied. We study the convolutors and the surjective Dunkl convolution operator acting on (Dαp)′, 1≤p≤∞. In the case p=2, we obtain complete characterizations.

We establish that imaginary powers of the Dunkl harmonic oscillator are bounded from L p /(ℝ d , dμ K ) to itself, 1 < p < ∞, and from L 1(ℝ d , dμ K ) to L 1,∞(ℝ d , dμ K ).

In the present paper, we establish that Riesz transforms for Dunkl Hermite expansions introduced by Nowak and Stempak are singular integral operators with Hormander's type condition. We prove that they are bounded on L p ( R d , d μ κ ) for 1 p ∞ and from L 1 ( R d , d μ κ ) into L 1 , ∞ ( R d , d μ κ ) .

We give the inverse of the Dunkl-type Segal–Bargmann transform Bα as a limit of certain integral operators. An application to Tikhonov regularization on the Dunkl-type Fock spaces Fα(C) is given using the theory of reproducing kernel Hilbert spaces.

The main purpose of this paper is to construct linear positive operators generated by Dunkl generalization of exponential function. We present qualitative result with the help of universal Korovkin-type theorem and weighted Korovkin-type theorem and also get quantitative estimates in terms of classical modulus of continuity, the class of Lipschitz functions, Peetre’s K-functional, second order modulus of continuity and weighted modulus of continuity by Dunkl analogue of Szasz operators.

Let f be the product of a homogeneous polynomial P on ℝd and a radial function. We calculate the Dunkl transform of f. When restricted to the classical case of the Fourier transform and P harmonic, the expression we get goes back to Bochner.

We study some class of Dunkl multiplier operators; and we establish for them some versions of uncertainty principles. For these operators we give also an application of the theory of reproducing kernels to the Tikhonov regularization on the Sobolev–Dunkl spaces.

In this paper, we present three characterizations of Dunkl-classical orthogonal polynomials. The first one is a -distributional equation of Pearson type fulfilled by its associated form, where is the Dunkl operator. The second is a first order linear -difference equation with polynomial coefficients satisfied by the corresponding Stieltjes function. The third is the so-called structure relation fulfilled by Dunkl–classical polynomials.

In this work, we establish a Babenko–Beckner-type inequality for the Dunkl transform of a radial function and Dunkl transform associated to a finite reflection group generated by the sign changes. As applications, we establish a new version of Young’s-type inequality and uncertainty relation for Renyi entropy.

This note is devoted to the intertwining operator in the one--dimensional trigonometric Dunkl setting. We obtain a simple integral expression of this operator and deduce its positivity., A para{\^i}tre dans Proc. Amer. Math. Soc

In this paper, we introduce new atomic Hardy-type spaces in the Dunkl setting and we investigate some properties of the Dunkl transform on . Next, we study the maximal Bochner–Riesz operators on these spaces.

In this article, we establish a characterization for the homogeneous Dunkl–Besov spaces via the Dunkl heat semigroup. Next we obtain the generalized Sobolev embedding theorems. Finally, we describe the Taylor series for the Dunkl Gaussian kernel.

We give an application of the theory of reproducing kernels to the Tikhonov regularization on the Sobolev–Dunkl spaces. Next, we come up with some results regarding the identity operator, the Dunkl transform and pseudo-differential operators.

In this paper, we give a generalization of the Heisenberg–Pauli–Weyl uncertainty inequality for the Dunkl transform on ℝ d .

This paper is devoted to the study of the domains of singularities in C (resp. in C × C ) of the operator valued functions k ↦ V k and ( k , k ′ ) ↦ V k , k ′ , where V k and V k , k ′ are the operators which intertwine the derivative operator d d x with respectively the Dunkl operator T ( k ) f ( x ) = d f d x ( x ) + k f ( x ) − f ( − x ) x and the Jacobi–Cherednik operator T ( k , k ′ ) f ( x ) = f ′ ( x ) + ( k coth ( x ) + k ′ tanh ( x ) ) ( f ( x ) − f ( − x ) ) − ( k + k ′ ) f ( − x ) . We also determine the singularities of the inverses and the duals of these operators V k and V k , k ′ by analytic methods and we show that some of them are entire functions whereas others are only meromorphic functions on C and C × C respectively.

This paper deals with a new singular differential-difference operator Y t, A on the real line including, as a particular case, the Dunkl, Dunkl–Heckman and Dunkl–Cherednik operators. We establish some results of Harmonic analysis related to this operator, such that a product formula for the related eigenfunction G λ is given as integral with an explicit kernel. This product formula is an important tool to define the generalized translation operator which is used to set up a convolution structure. Next, we establish an inversion formula and prove a generalized Plancherel theorem for this operator. As a direct application, we give a maximum principle of the operators and we solve the heat equation associated with the generalized Dunkl operator on the real line.

We introduce moduli of continuity on the basis of the generalized translation generated by the Dunkl operators $$ \Lambda f(x) = f^{\prime}(x) + \frac{{2\alpha + 1}}{x}\frac{{f(x) - f\left( { - x} \right)}}{2} $$ and estimate the functionals in terms of these moduli. The constants in these inequalities are explicitly expressed in terms of the moments of functionals, i.e. their upper bounds on generalized Sobolev classes. Jackson-type inequalities in the Lebesgue spaces on the real line with the weight $ {A_\alpha }(x) = {\left| x \right|^{2\alpha + 1}} $ , $ \alpha > - \frac{1}{2} $ , are particular cases of the inequalities established. Bibliography: 27 titles.

In this paper, we study the Dunkl–Bessel potential associated with the Dunkl Laplacian and give some of its properties. Next, using its expression in terms of the Dunkl metaharmonic semigroup, we determine its inverse on , 1≤p

In this paper, we study a family of generalized windowed transforms associated with the Dunkl operators on ℝ d , and we prove for them Plancherel and inversion formulas and characterize their images.

In this paper we present an approach to obtain harmonic function in the higher dimensional spaces in Dunkl-Clifford analysis. A very important application of this method is to establish some counterparts of Fueter’s Theorem in Dunkl case. To this end the corresponding Vekua-type systems for poly-axial Dunkl-monogenic functions are considered and studied.

In this note, Kelvin transform is introduced in the framework of Dunkl-Clifford analysis. It is shown that this transform preserves the class of Dunkl monogenic functions. As an application, we use it to generate Dunkl monogenic polynomials by a classical process due to Maxwell.

Using a generalized spherical mean operator, we obtain an analog of Titchmarsh's Theorem for the Dunkl transform for functions satisfying the spherical Lipschitz condition in L 2(ℝ d , ω k ), where ω k is a weight function invariant under the action of an associated Weyl group.

WOS: 000280106600007 On the real line, the Dunkl operators are differential-difference operators associated with the reflection group Z(2) on R. In this paper, we obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional maximal operator associated with the Dunkl operator on R from the spaces L-p,L-alpha(R) to the spaces L-q,L-alpha(R) and from the spaces L-1,L-alpha(R) to the weak spaces W L-q,L-alpha(R). Azerbaijan-U.S. Bilateral Grants Program IIAzerbaijan National Science Foundation (ANSF) [16071] V. Guliyev's research was partially supported by the grant of the Azerbaijan-U.S. Bilateral Grants Program II (project ANSF Award/16071).

WOS: 000280762600001, We consider the generalized shift operator, associated with the Dunkl operator Lambda(alpha)(f)(x) = (d/dx)f(x) + ((2 alpha + 1)/x)((f(x) - f(-x))/2), alpha > -1/2. We study the boundedness of the Dunkltype fractional maximal operator M(beta) in the Dunkl-type Morrey space L(p,lambda,alpha)(R), 0

We consider on ℝ the differential-difference operator which is referred to as the Dunkl operator with parameter γ+1/2 associated with the reflection ℤ2 on ℝ. We exhibit a Sonine-type integral representation for the Dunkl kernel e γ, eigenfunction of the Dunkl operator Λγ. This enables us to construct a new Sonine integral transform on ℝ tied to Λγ, which turns out together with its dual to be transmutation operators between two Dunkl operators of different parameters.