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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. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present some recent results for maximal functions and to study the $$L^p$$ -boundedness of the spherical maximal function associated to the Dunkl operators. [ABSTRACT FROM AUTHOR]

In this paper, we establish new spectral theorems for the Dunkl transform on $$\mathbb {R}^{d}$$ . [ABSTRACT FROM AUTHOR]

We introduce the notion of Dunkl positive definite and strictly pos- itive definite functions on ℝd. This done by the use of the properties of Dunkl translation. We establish the analogue of Bochner's theorem in Dunkl setting. The case of radial functions is considered. We give a sufficient condition for a function to be Dunkl strictly positive definite on ℝd. [ABSTRACT FROM AUTHOR]

This paper deals with generalized Lipschitz spaces $$\wedge ^k_{\alpha ,p,q}(\mathbb {R})$$ in the context of Dunkl harmonic analysis on $$\mathbb {R}$$ , for all real $$\alpha $$ . It also introduces a generalized Dunkl-Lipschitz spaces $$\mathcal{T}\wedge ^k_{\alpha ,p,q}(\mathbb {R}^2_+)$$ of k-temperature on $$\mathbb {R}^2_+$$ . Some properties and continuous embedding of these spaces and the isomorphism of $$\mathcal{T}\wedge ^k_{\alpha ,p,q}(\mathbb {R}^2_+)$$ and $$\wedge ^k_{\alpha ,p,q}(\mathbb {R})$$ are established. [ABSTRACT FROM AUTHOR]

The Heisenberg uncertainty principle and the uncertainty principle for self-adjoint operators have been known and applied for decades. In this paper, in the framework of Clifford algebra, we establish a stronger Heisenberg-Pauli-Wely type uncertainty principle for the Fourier transform of multivector-valued functions, which generalizes the recent results about uncertainty principles of Clifford-Fourier transform. At the end, we consider another stronger uncertainty principle for the Dunkl transform of multivector-valued functions. [ABSTRACT FROM AUTHOR]

For a root system in Rd furnished with its Coxeter-Weyl group W and a multiplicity nonnegative function k, we consider the associated commuting system of Dunkl operators D1, . . ., Dd and the Dunkl-Laplacian Δk = D1²+. . .+Dd². This paper studies the properties of the functions u defined on an open W-invariant set Ω ⊂ Rd and satisfying Δku = 0 on Ω (D-harmonicity). In particular, we introduce and give a complete study of a new mean value operator which characterizes D-harmonicity. As applications we prove a strong maximum principle, a Harnack's type theorem and a Bôcher's theorem for D-harmonic functions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We prove sharp Pitt’s inequality for the Dunkl transform in L 2 ( R d ) with the corresponding weights. As an application, we obtain the logarithmic uncertainty principle for the Dunkl transform. [ABSTRACT FROM AUTHOR]

In this article, we prove An Lp-Lq-version of Morgan's theorem for the generalized Dunkl transform. [ABSTRACT FROM AUTHOR]

In the present paper, we investigate in Dunkl analysis, the action of some fundamental operators on the atomic Hardy space $$H^1_k$$ . [ABSTRACT FROM AUTHOR]

We study the boundedness of the multiplier of the intervalfor the Dunkl transform of orderon weightedspaces, with. In particular, we get that it is bounded frominto itself if and only ifwhenor if and only ifwhen. [ABSTRACT FROM AUTHOR]

The Dunkl transform ${\mathcal{F}}_{k}$ is a generalisation of the usual Fourier transform to an integral transform invariant under a finite reflection group. The goal of this paper is to prove a strong uncertainty principle for orthonormal bases in the Dunkl setting which states that the product of generalised dispersions cannot be bounded for an orthonormal basis. Moreover, we obtain a quantitative version of Shapiro’s uncertainty principle on the time–frequency concentration of orthonormal sequences and show, in particular, that if the elements of an orthonormal sequence and their Dunkl transforms have uniformly bounded dispersions then the sequence is finite. [ABSTRACT FROM PUBLISHER]

Using a generalized spherical mean operator, we obtain an analog of Theorem 5.2 in Younis (J Math Sci 9(2),301-312 ) for the Dunkl transform for functions satisfying the $$d$$ -Dunkl Dini Lipschitz condition in the space $$\mathrm {L}^{2}(\mathbb {R}^{d},w_{k}(x)dx)$$ , where $$w_{k}$$ is a weight function invariant under the action of an associated reflection group. [ABSTRACT FROM AUTHOR]

In the case , we prove a multivariate Marcinkiewicz-type multiplier theorem for the Dunkl transform and a multivariate Marcinkiewicz-type multiplier theorem for the system of Dunkl harmonic oscillators. [ABSTRACT FROM AUTHOR]

In the present paper, we study first the behavior at infinity of the Riesz potential for Dunkl transform of a non compactly supported function. Second, we give for $$1 < p \le q < +\infty $$ , weighted $$Lp\rightarrow Lq$$ boundedness of the Riesz potentials with sufficient conditions. As application, we prove a weighted generalized Sobolev inequality. [ABSTRACT FROM AUTHOR]

Letfbe the product of a homogeneous polynomialPon ?dand a radial function. We calculate the Dunkl transform off. When restricted to the classical case of the Fourier transform andPharmonic, the expression we get goes back to Bochner. [ABSTRACT FROM AUTHOR]

The paper is devoted to the determination of the optimal arguments in the sharp Jackson-Stechkin inequality with modulus of continuity of order r in the space L(ℝ) with Dunkl weight defined by the root system R and a nonnegative function of multiplicity k. If where ℝ is the positive subsystem of the root system, then the optimal arguments for all r coincide. If λ ≠ 1/2, then the optimal argument for the modulus of continuity of second order is greater than for the first order. Such patterns are related to the arithmetic properties of zeros of Bessel functions. [ABSTRACT FROM AUTHOR]

In this paper, we study the sharp Jackson inequality for the best approximation of f ∈ L(ℝ) by a subspace E( σ) ( SE( σ)), which is a subspace of entire functions of exponential type (spherical exponential type) at most σ. Here L(ℝ) denotes the space of all d-variate functions f endowed with the L-norm with the weight $v_\kappa (x) = \prod\nolimits_{\xi \in R_ + } {|(\xi ,x)|^{2\kappa (\xi )} } $, which is defined by a positive subsystem R of a finite root system R ⊂ ℝ and a function κ( ξ): R → ℝ invariant under the reflection group G( R) generated by R. In the case G( R) = ℤ, we get some exact results. Moreover, the deviation of best approximation by the subspace E( σ) ( SE( σ)) of some class of the smooth functions in the space L(ℝ) is obtained. [ABSTRACT FROM AUTHOR]

We show Stein–Weiss inequality and Hardy–Littlewood–Sobolev inequality in the Dunkl setting; and we deduce some versions of Pitt's inequality for the Dunkl transform on ℝd. [ABSTRACT FROM AUTHOR]

We establish Pitt's inequality and deduce Beckner's logarithmic uncertainty principle for the Dunkl transform on $${\mathbb{R}}$$ . Also, we prove Stein-Weiss inequality for the Dunkl-Bessel potentials. [ABSTRACT FROM AUTHOR]

A new fractional version of the Dunkl transform for real order is obtained. An integral representation, a Bochner type identity and a Master formula for this transform are derived. [ABSTRACT FROM AUTHOR]

In this work we consider the Dunkl operator on the real line, defined by We define and study Dunkl-Sobolev spaces $L^p_{n,k}(\mathbb{R})$, Dunkl-Sobolev spaces ${\cal L}^p_{\alpha,k}(\mathbb{R})$ of positive fractional order and generalized Dunkl-Lipschitz spaces $\wedge^k_{\alpha,p,q}(\mathbb{R})$. We provide characterizations of these spaces and we give some connection between them. [ABSTRACT FROM AUTHOR]

Two estimates useful in applications are proved for the Dunkl transform in Lp,α = Lp (ℝ, |x|2α+1 dx) as applied to some classes of functions characterized by a generalized modulus of continuity. [ABSTRACT FROM AUTHOR]

Abstract: We consider several harmonic analysis operators in the multi-dimensional context of the Dunkl Laplacian with the underlying group of reflections isomorphic to (also negative values of the multiplicity function are admitted). Our investigations include maximal operators, g-functions, Lusin area integrals, Riesz transforms and multipliers of Laplace and Laplace–Stieltjes transform type. Using the general Calderón–Zygmund theory we prove that these objects are bounded in weighted spaces, , and from into weak . [Copyright &y& Elsevier]

Using a generalized spherical mean operator, we obtain a generalization of Titchmarsh's theorem for the Dunkl transform for functions satisfying the (ψ,α,β)-Dunkl Lipschitz condition in L² (Rd, wk(x)dx). [ABSTRACT FROM AUTHOR]

Using a generalized Dunkl translation, we obtain an analog of Theorem 5.2 in Younis' paper for the Dunkl transform for functions satisfying the (δ, γ)-Dunkl Lipschitz condition in the space L² (ℝ, |x|2α+1dx). [ABSTRACT FROM AUTHOR]

In Dunkl theory on $$\mathbb R ^d$$ which generalizes classical Fourier analysis, we prove first weighted inequalities for certain Hardy-type averaging operators. In particular, we deduce for specific choices of the weights the $$d$$-dimensional Hardy inequalities whose constants are sharp and independent of $$d$$. Second, we use the weight characterization of the Hardy operator to prove weighted Dunkl transform inequalities. As consequence, we obtain Pitt's inequality which gives an integrability theorem for this transform on radial Besov spaces. [ABSTRACT FROM AUTHOR]

This paper is devoted to the determination of the optimal arguments in the exact Jackson inequality in the space L on the Euclidean space with power weight equal to the product of the moduli of the coordinates with nonnegative powers. The optimal arguments are studied depending on the geometry of the spectrum of the approximating entire functions and the neighborhood of zero in the definition of the modulus of continuity. The optimal arguments are obtained in the case where the first skew field is a l-ball for 1 ≤ p≤ 2, and the second is a parallelepiped. [ABSTRACT FROM AUTHOR]

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

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. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove new uncertainty principles for the Dunkl transform. More precisely, we show a variation on Heisenberg's uncertainty inequality, the local uncertainty principle and Donoho–Stark's uncertainty principle. [ABSTRACT FROM AUTHOR]

In this paper, we give a generalization of the Heisenberg–Pauli–Weyl uncertainty inequality for the Dunkl transform on ℝd. [ABSTRACT FROM AUTHOR]

This paper presents a systematic study for harmonic analysis associated with the one-dimensional Dunkl transform, which is based upon the generalized Cauchy-Riemann equations D u− ∂ v=0, ∂ u+ D v=0, where D is the Dunkl operator ( D f)( x)= f′( x)+( λ/ x)( f( x)− f(− x)). Various properties about the λ-subharmonic function, the λ-Poisson integral, the conjugate λ-Poisson integral, and the associated maximal functions are obtained, and the λ-Hilbert transform [InlineEquation not available: see fulltext.], a crucial analog to the classical one, is introduced and studied by a stringent method. The theory of the associated Hardy spaces $H_{\lambda}^{p}({\mathbb{R}}^{2}_{+})$ on the half-plane ${\mathbb{R}}^{2}_{+}$ for p≥ p=2 λ/(2 λ+1) with λ>0 extends the results of Muckenhoupt and Stein about the Hankel transform to a general case and contains a number of further results. In particular, the λ-Hilbert transform [InlineEquation not available: see fulltext.] is shown to be a bounded mapping from $H_{\lambda}^{1}({\mathbb{R}})$ to $L^{1}_{\lambda}({\mathbb{R}})$; and associated to the Dunkl transform, an analog of the well-known Hardy inequality is proved for $f\in H^{1}_{\lambda}({\mathbb{R}})$. [ABSTRACT FROM AUTHOR]

In this paper, we give analogues of the local uncertainty inequality for the Dunkl transform on ℝd, and indicate how the local uncertainty inequality implies a global uncertainty inequality. [ABSTRACT FROM AUTHOR]

Abstract: We prove that the system ( is Dunkl kernel) is a Riesz basis in corresponding space if is the sequence of zeros of an entire function from certain class. [Copyright &y& Elsevier]

Abstract: We obtain a q-linear analogue of Gegenbauerʼs expansion of the plane wave. It is expanded in terms of the little q-Gegenbauer polynomials and the third Jackson q-Bessel function. The result is obtained by using a method based on bilinear biorthogonal expansions. [Copyright &y& Elsevier]

In this review, we give an overview of several recent generalizations of the Fourier transform, related to either the Lie algebra [ABSTRACT FROM AUTHOR]

We introduce first weighted function spaces on $${\mathbb{R}^d}$$ using the Dunkl convolution that we call Besov-Dunkl spaces. We provide characterizations of these spaces by decomposition of functions. Next we obtain in the real line and in radial case on $${\mathbb{R}^d}$$ weighted L-estimates of the Dunkl transform of a function in terms of an integral modulus of continuity which gives a quantitative form of the Riemann-Lebesgue lemma. Finally, we show in both cases that the Dunkl transform of a function is in L when this function belongs to a suitable Besov-Dunkl space. [ABSTRACT FROM AUTHOR]

Abstract: Given , and , we study the interpolation problem in the space of entire functions of exponential type ⩽σ for which , with nodes of interpolation at , , where is the increasing sequence of all positive roots of the Bessel function of order , and for all . We prove that if , the interpolation problem has a unique solution for every sequence of complex numbers satisfying , and that if , the corresponding interpolation problem may not have a solution, and that the solution, if exists, is unique if and only if . Finally, we show that with the constant of equivalence depending only on p and α, holds for all entire functions f of exponential type ⩽σ if and only if . [Copyright &y& Elsevier]

In the spaces L on the line with power weight, we study approximation of functions by entire functions of exponential type. Using the Dunkl difference-differential operator and the Dunkl transform, we define the generalized shift operator, the modulus of smoothness, and the K-functional. We prove a direct and an inverse theorem of Jackson-Stechkin type and of Bernstein type. We establish the equivalence between the modulus of smoothness and the K-functional. [ABSTRACT FROM AUTHOR]