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In this work, we define the composition of wavelet transforms and obtain its Parsevals's identity. Furthermore, we discuss the convolution operator and continuous Dunkl wavelet transform as time-invariant filters. The physical interpretation and potential application of time-invariant filter involving Fredholm type integral are obtained. [ABSTRACT FROM AUTHOR]

The purpose of this article is to present one and two-weight inequalities for bilinear multiplier operators in Dunkl setting with multiple Muckenhoupt weights. In order to do so, new results regarding Littlewood–Paley type theorems and weighted inequalities for multilinear Calderón–Zygmund operators in Dunkl setting are also proved. [ABSTRACT FROM AUTHOR]

We derive asymptotic estimates for the growth of the norm of the deformed Hankel transform on the deformed Hankel–Lipschitz space defined via a generalised modulus of continuity. The established results are similar in nature to the well-known Titchmarsh theorem, which provide a characterization of the square integrable functions satisfying certain Cauchy–Lipschitz condition in terms of an asymptotic estimate for the growth of the norm of their Fourier transform. We also give some necessary conditions in terms of the generalised modulus of continuity for the boundedness of the Dunkl transform of functions in Dunkl–Lipschitz spaces, improving the Hausdorff–Young inequality for the Dunkl transform in this special scenario. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to show that the modulus of smoothness and the $K$-functionals constructed from the Sobolev-type space corresponding to the Dunkl operator are equivalent on the interval $(-1,1)$.

In this paper, utilizing the Dunkl transform and a generalized translation operator, we derive an analogue of the Riesz-Feller fractional derivative for a class of functions satisfying Lipschitz conditions. [ABSTRACT FROM AUTHOR]

In this paper, we define the Weyl transform associated with Dunkl wavelet transform and discuss its boundedness and compactness on the Lebesgue spaces. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study for the Dunkl–Stockwell transform some problems of the reproducing kernels theory. In particular, we investigate the Saitoh approach to give some applications of the Tikhonov regularization for some generalized Sobolev spaces associated with the Dunkl–Stockwell transform and we study also some time-frequency concentration problems. [ABSTRACT FROM AUTHOR]

On R N equipped with a root system R and a multiplicity function k > 0 , we study the generalized (Dunkl) translations τ x g (- y) of not necessarily radial kernels g. Under certain regularity assumptions on g, we derive bounds for τ x g (- y) by means the Euclidean distance ‖ x - y ‖ and the distance d (x , y) = min σ ∈ G ‖ x - σ (y) ‖ , where G is the reflection group associated with R. Moreover, we prove that τ does not preserve positivity, that is, there is a non-negative Schwartz class function φ , such that τ x φ (- y) < 0 for some points x , y ∈ R N . [ABSTRACT FROM AUTHOR]

In this paper, we study non–local in time evolution type equations generated by the Dunkl operator. Direct and inverse problems are investigated with the Caputo time-fractional heat equation with the parameter 0 < γ ≤ 1 . In particular, well-posedness properties are established for the forward problem. To adopt techniques of the harmonic analysis, we solve the problems in the Sobolev type spaces associated with the Dunkl operator. Our special interest is an inverse source problem for the Caputo–Dunkl heat equation. As additional data, the final time measurement is taken. Since our inverse source problem is ill-posed, we also show the stability result. Moreover, as an advantage of our calculus used here, we derive explicit formulas for the solutions of the direct and inverse problems. [ABSTRACT FROM AUTHOR]

The deformed wavelet transform is a new addition to the class of wavelet transforms that heavily rely on a pair of generalized translation and dilation operators governed by the well-known Dunkl transform. In this study, we adapt the symmetrical properties of the Dunkl Laplacian operator to prove a class of quantitative uncertainty principles associated with the deformed wavelet transform, including Heisenberg's uncertainty principle, the Benedick–Amrein–Berthier uncertainty principle, and the logarithmic uncertainty inequalities. Moreover, using the symmetry between a square integrable function and its Dunkl transform, we establish certain local-type uncertainty principles involving the mean dispersion theorems for the deformed wavelet transform. [ABSTRACT FROM AUTHOR]

Some examples of Appell–Dunkl sequences are shown using determined operators. Specifically, Appell–Dunkl sequences whose generating functions are of the form E α (x t) / (1 ± t m) , where the function E α (x t) is given in terms of Bessel functions. Particular cases of these examples are also generated by means of the inverse of the Dunkl operator. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce the Dunkl—Hilbert transform Hk, with k ≥ 0, induced by the Dunkl differential operator and associated with the reflection group ℤ2. For this end, we establish that the Dunkl—Poisson kernel and the conjugate Dunkl—Poisson kernel satisfy the Cauchy—Riemann equations in the Dunkl context. We prove the continuity of Hk on Lp(wk) for 1 < p < ∞, where wk(x) = ∣x∣2k. Finally, we introduce the maximal Hilbert operator H*k and establish an analogue of Cotlar's theorem. [ABSTRACT FROM AUTHOR]

In this paper we study the two-sided quaternionic Dunkl transform. We establish its fundamental properties, such as Plancherel and inversion formula. Finally, we apply the Dunkl transform properties to establish an analogue of Hardy's theorem [ABSTRACT FROM AUTHOR]

Our aim in this paper is to showthat the modulus of smoothness and the K-functionals constructed from the Sobolev-type space corresponding to the Dunkl operator are equivalent on the interval (-1, 1). [ABSTRACT FROM AUTHOR]

In this paper we define multilinear fractional integral operator and multilinear fractional maximal function in Dunkl setting and present a weighted theory for these operators. Sufficient conditions for two-weight inequalities are found using “power-bump” conditions and for one-weight inequalities Muckenhoupt-type conditions are obtained as sufficient condition. [ABSTRACT FROM AUTHOR]

By using the symmetry of the Dunkl Laplacian operator, we prove a sharp Shannon-type inequality and a logarithmic Sobolev inequality for the Dunkl transform. Combining these inequalities, we obtain a new, short proof for Heisenberg-type uncertainty principles in the Dunkl setting. Moreover, by combining Nash's inequality, Carlson's inequality and Sobolev's embedding theorems for the Dunkl transform, we prove new uncertainty inequalities involving the L ∞ -norm. Finally, we obtain a logarithmic Sobolev inequality in L p -spaces, from which we derive an L p -Heisenberg-type uncertainty inequality and an L p -Nash-type inequality for the Dunkl transform. [ABSTRACT FROM AUTHOR]

A famous result of Beurling says that if the Fourier transform of a non-zero integrable function on the real line has certain exponential decay, then the function cannot vanish on a set of positive Lebesgue measure. We prove an analogue of Beurling's theorem for Hankel transform and some several variable analogues of Beurling's theorem for Fourier transform as a consequence of similar results for Dunkl transform. We also prove an analogue of Beurling's theorem for spectral projections associated to the Dunkl-Laplacian. [ABSTRACT FROM AUTHOR]

The deformed wavelet transform is a new addition to the class of wavelet transforms that heavily rely on a pair of generalized translation and dilation operators governed by the well-known Dunkl transform. In this study, we adapt the symmetrical properties of the Dunkl Laplacian operator to prove a class of quantitative uncertainty principles associated with the deformed wavelet transform, including Heisenberg’s uncertainty principle, the Benedick–Amrein–Berthier uncertainty principle, and the logarithmic uncertainty inequalities. Moreover, using the symmetry between a square integrable function and its Dunkl transform, we establish certain local-type uncertainty principles involving the mean dispersion theorems for the deformed wavelet transform.

Let G be a noncompact semisimple Lie group with finite centre. Let X = G / K be the associated Riemannian symmetric space and assume that X is of rank one. The generalized spectral projections associated to the Laplace-Beltrami operator are given by P λ f = f ∗ Φ λ , where Φ λ are the elementary spherical functions on X. In this paper, we prove an Ingham type uncertainty principle for P λ f . Moreover, similar results are obtained in the case of generalized spectral projections associated to Dunkl Laplacian. [ABSTRACT FROM AUTHOR]

In this paper, we strengthen the results of the uncertainty principle established by M. Rösler and M. Voit and the real Paley–Wiener theorem established by N. B. Anderson and M. de Jeu for the Dunkl transform on the real line. Examples are provided to show that the new uncertainty principle is truly sharper than the existing one in literature. [ABSTRACT FROM AUTHOR]

We study weighted (Lp, Lq)-boundedness properties of Riesz potentials and fractional maximal functions for the Dunkl transform. In particular, we obtain the weighted Hardy–Littlewood–Sobolev type inequality and weighted week (L1, Lq) estimate. We find a sharp constant in the weighted Lp-inequality, generalizing the results of W. Beckner and S. Samko. [ABSTRACT FROM AUTHOR]

After reviewing Dunkl Pitt's and Dunkl Beckner's inequalities we connect both the inequalities to show a generalization of uncertainty principles for the Dunkl Gabor transform. Next we present two concentration uncertainty principles such as Benedick-Amrein-Berthier's uncertainty principle and local uncertainty principle. Finally, we study the Dunkl logarithmic Sobolev inequalities. Obtaining best possible constants of inequalities, we connect the inequalities to show a generalization of the uncertainty principles of Heisenberg type. [ABSTRACT FROM AUTHOR]

In this paper, we extend the one-dimensional Stockwell transform discussed in Stockwell et al. (IEEE Trans Signal Process 44:998–1001, 1996) to the Dunkl theory seeting. We obtain the expected properties of the extended Stockwell transform such as Parseval’s formula, inversion formula, Calderón’s formula. Next, some applications are given as the uncertainty principles and the theory of localization operators associated with this new transform. [ABSTRACT FROM AUTHOR]

In this work, we prove Clarkson-type and Nash-type inequalities in the Dunkl setting on ℝ d for L p -functions. By combining these inequalities, we show Heisenberg-type inequalities for the Dunkl transform on ℝ d , and we deduce local-type uncertainty inequalities for the Dunkl transform on ℝ d . [ABSTRACT FROM AUTHOR]

By using the symmetry of the Dunkl Laplacian operator, we prove a sharp Shannon-type inequality and a logarithmic Sobolev inequality for the Dunkl transform. Combining these inequalities, we obtain a new, short proof for Heisenberg-type uncertainty principles in the Dunkl setting. Moreover, by combining Nash’s inequality, Carlson’s inequality and Sobolev’s embedding theorems for the Dunkl transform, we prove new uncertainty inequalities involving the L∞-norm. Finally, we obtain a logarithmic Sobolev inequality in Lp-spaces, from which we derive an Lp-Heisenberg-type uncertainty inequality and an Lp-Nash-type inequality for the Dunkl transform.

In this paper, we prove the existence of an extremal for the Dunkl-type Sobolev inequality in the case of p = 2 . Also we prove the existence of an extremal of the Stein–Weiss inequality for the Dunkl Riesz potential in the case of r = 2 . [ABSTRACT FROM AUTHOR]

We extend Hardy's theorem for rotation due to Lakey and Hogan for Dunkl transform associated with the reflection group Z 2 in R by using Dunkl-Hermite semigroup. In fact, we characterize f ∈ L 1 ∩ L 2 (R , h κ 2) by considering the decay of f and it's Dunkl transform F κ f along some lines in the complex plane, and we give a d-dimensional analogue for the same corresponding to the reflection group Z 2 d. Moreover, we prove a version of Hardy's theorem, assuming the decay of Dunkl-Hermite coefficients. [ABSTRACT FROM AUTHOR]

We prove two theorems showing connections between multipliers for the symmetrized Jacobi expansions and multipliers for the Dunkl transform on R . These results can be seen as generalizations of the classical Igari's result relating Jacobi and Hankel multipliers. Moreover, we show a transference theorem relating Jacobi and Dunkl transplantations. [ABSTRACT FROM AUTHOR]

In this paper we study certain relations between the Dunkl–Sobolev classes of fractional order Lα,kp(IR) and the generalized Dunkl–Lipschitz classes ∧α,p,qk(IR). Next, we show a Dunkl–Hardy–Littlewood theorem for k‐temperatures and its dual. Also, we give some theorems on Dunkl convolutions of Dunkl–Lipschitz functions. Finally, we establish some results concerning the Dunkl transforms of functions and distributions belonging to generalized Dunkl–Lipschitz spaces. [ABSTRACT FROM AUTHOR]

We define a fractional power of the Dunkl Laplacian, a fractional modulus of smoothness, and a fractional K-functional on Lp-spaces with Dunkl weight. As an application, we extend our previous results and prove direct and inverse theorems of approximation theory and some inequalities for entire functions of spherical exponential type in the fractional setting. [ABSTRACT FROM AUTHOR]

We prove a partial fraction decomposition of a quotient of two functions E α (i t x) and I α (i t) which are defined in terms of the Bessel functions J α and J α + 1 of the first kind. This expansion leads naturally to the introduction of an orthonormal system with respect to the measure | x | 2 α + 1 d x 2 α + 1 Γ (α + 1) in [ − 1 , 1 ] , which we call the Fourier–Dunkl system of the second kind. Euler–Dunkl polynomials E n , α (x) of degree n are defined by considering E α (t x) ∕ I α (t) as a generating function. It is shown that the sum ∑ m = 1 ∞ 1 ∕ j m , α 2 k , where j m , α are the positive zeros of J α , is equal (up to an explicit factor) to E 2 k − 1 , α (1). For α = 1 ∕ 2 this leads to classical results of Euler since the function E 1 ∕ 2 (x) is the exponential function and E n , 1 ∕ 2 (x) are (essentially) the Euler polynomials. In the second part of the paper a sampling theorem of Whittaker–Shannon–Kotel'nikov type is established which is strongly related to the above-mentioned partial decomposition and which holds for all functions in the Payley–Wiener space defined by the Dunkl transform in [ − 1 , 1 ]. [ABSTRACT FROM AUTHOR]

Using a generalized spherical mean operator, we obtain a generalization of two theorems 84 and 85 of Titchmarsh for the Dunkl transform for functions satisfying the Dunkl—Lipschitz condition in the space Lp(ℝd, wk(x)dx), where 1 < p ≤ 2. [ABSTRACT FROM AUTHOR]

We prove that the spherical mean value of the Dunkl-type generalized translation operator τ y is a positive L p -bounded generalized translation operator T t . As applications, we prove the Young inequality for a convolution defined by T t , the L p -boundedness of τ y on radial functions for p > 2 , the L p -boundedness of the Riesz potential for the Dunkl transform, and direct and inverse theorems of approximation theory in L p -spaces with the Dunkl weight. [ABSTRACT FROM AUTHOR]

A sharp Jackson inequality in the space Lp(ℝd), 1 ≤ p < 2, with Dunkl weight is proved. The best approximation is realized by entire functions of exponential spherical type. The modulus of continuity is defined by means of a generalized shift operator bounded on Lp, which was constructed earlier by the authors. In the case of the unit weight, this operator coincides with the mean-value operator on the sphere. [ABSTRACT FROM AUTHOR]

In the paper we study the Schrödinger oscillatory integrals Tλ,atf(x) (λ≥0, a>1) associated with the one-dimensional Dunkl transform Fλ. If a=2, the function u(x,t):=Tλ,2tf(x) solves the free Schrödinger equation associated to the Dunkl operator, with f as the initial data. It is proved that, if f is in the Sobolev spaces Hλs(R) associated with the Dunkl transform, with the exponents s not less than 1 / 4, then Tλ,atf converges almost everywhere to f as t→0. A counterexample is constructed to show that 1 / 4 can not be improved for a=2, and when 1/4≤s≤1/2, the Hausdorff dimension of the divergence set of Tλ,atf for f∈Hλs(R) is proved to be 1-2s at most. [ABSTRACT FROM AUTHOR]

In the present paper, we prove weighted inequalities for the Dunkl transform (which generalizes the Fourier transform) when the weights belong to the well-known class Bp. As application, we obtain the Pitt’s inequality for power weights

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 . 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 , p>1, and from to weak . [ABSTRACT FROM AUTHOR]

Using a generalized Dunkl translation, we obtain an analog of Theorem 5.2 in Younis' paper [2] for the Dunkl transform for functions satisfying the $(\delta, \gamma)$-Dunkl Lipschitz condition in the space $\mathrm{L}^{2}(\mathbb{R}, |x|^{2\alpha+1}dx)$.}

We introduce the notion of a Dunkl two-wavelet multiplier, and we give its trace formula as a bounded linear operator in the trace class from into in terms of the symbol and the two admissible wavelets. Next, we give results on the boundedness and compactness of Dunkl two-wavelet multipliers on , . [ABSTRACT FROM AUTHOR]

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. Calderón'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. [ABSTRACT FROM AUTHOR]

A Littlewood–Paley operator associated with the reflection part of the Dunkl operator is introduced and proved to be of type $(p,p)$ for $1

We give a solution of the Bohman extremal problem for nonnegative functions with the support of the Dunkl transform in a Euclidean ball or a parallelepiped. The proof uses the invariance of the problem under orthogonal transforms and quadrature formulas with zeros of Bessel functions as nodes. [ABSTRACT FROM AUTHOR]

Using a generalized spherical mean operator, we obtain a generalization of Younis's Theorem 5.2 in [12] for the Dunkl transform for functions satisfying the d-Dunkl Dini Lipschitz condition in the space Lp(Rd;ωl(x)dx), 1 < p ≤ 2, where ωl is a weight function invariant under the action of an associated reection group. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the direct theorem of approximation of functions on the real line $$\mathbb {R}$$ in the metric of $$L_{p}$$ with some power weight using generalized Dunkl shifts. [ABSTRACT FROM AUTHOR]