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1.
In this paper we consider the Dunkl operators T j , j = 1, . . . , d, on and the harmonic analysis associated with these operators. We define a continuous Dunkl Gabor transform, involving the Dunkl translation operator, by proceeding as mentioned in [20] by C.Wojciech and G. Gigante. We prove a Plancherel formula, an inversion formula and a weak uncertainty principle for it. Then, we show that the portion of the continuous Dunkl Gabor transform lying outside some set of finite measure cannot be arbitrarily too small. Similarly, using the basic theory for the Dunkl continuous wavelet transform introduced by K. Trimèche in [18], an analogous of this result for the Dunkl continuous wavelet transform is given. Finally, an analogous of Heisenberg’s inequality for a continuous Dunkl Gabor transform (resp. Dunkl continuous wavelet transform) is proved.   相似文献   

2.
By expressing the Dunkl transform of order α of a function f in terms of the Hankel transforms of orders α and α + 1 of even and odd parts of f, respectively, we show that a considerable part of harmonic analysis of the Dunkl transform on the real line may be reduced to known results for the Hankel transform. In particular, defining the modified Dunkl transform and then considering the Dunkl transplantation operator we transfer known multiplier results for the Hankel transform to the Dunkl transform setting. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

3.
In the spaces L p 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.  相似文献   

4.
The main purpose of this article is to study the L p -boundedness of linear and bilinear multiplier operators for the Dunkl transform in the one dimensional case.  相似文献   

5.
For a family of weight functionsh K invariant under a finite reflection group onR d, analysis related to the Dunkl transform is carried out for the weightedL p spaces. Making use of the generalized translation operator and the weighted convolution, we study the summability of the inverse Dunkl transform, including as examples the Poisson integrals and the Bochner-Riesz means. We also define a maximal function and use it to prove the almost everywhere convergence. ST wishes to thank YX for the warm hospitality during his stay in Eugene. The work of YX was supported in part by the National Science Foundation under Grant DMS-0201669.  相似文献   

6.
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 p -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 1 when this function belongs to a suitable Besov-Dunkl space.  相似文献   

7.
We prove real Paley-Wiener type theorems for the Dunkl transform ℱ D on the space of tempered distributions. Let TS′(ℝ d ) and Δ κ the Dunkl Laplacian operator. First, we establish that the support of ℱ D (T) is included in the Euclidean ball , M>0, if and only if for all R>M we have lim  n→+∞ R −2n Δ κ n T=0 in S′(ℝ d ). Second, we prove that the support of ℱ D (T) is included in ℝ d ∖B(0,M), M>0, if and only if for all R<M, we have lim  n→+∞ R 2n  ℱ D −1(‖y−2n D (T))=0 in S′(ℝ d ). Finally, we study real Paley-Wiener theorems associated with -slowly increasing function.   相似文献   

8.

Let Ω be a G-invariant convex domain in ℝN including 0, where G is a Coxeter group associated with reduced root system R. We consider functions f defined in Ω which are Dunkl polyharmonic, i.e. (Δh)nf = 0 for some integer n. Here333-01is the Dunkl Laplacian, and Dj is the Dunkl operator attached to the Coxeter group G,

$$\mathcal{D}_j f(x) = \frac{\partial }{{\partial x_j }}f(x) + \sum\limits_{v \in R_ + } {\kappa _v \frac{{f(x) - f(\sigma _v x)}}{{\left\langle {x,v} \right\rangle }}} v_j ,$$

where Kv is a multiplicity function on R and σv is the reflection with respect to the root v. We prove that any Dunkl polyharmonic function f has a decomposition of the form

$$f(x) = f_0 (x) + \left| x \right|^2 f_1 (x) + \cdots + \left| x \right|^{2(n - 1)} f_{n - 1} (x), \forall x \in \Omega ,$$

where fj are Dunkl harmonic functions, i.e. Δhfj = 0. This generalizes the classical Almansi theorem for polyharmonic functions as well as the Fischer decomposition.

  相似文献   

9.
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 x u?? y v=0,? y u+D x v=0, where D x is the Dunkl operator (D x 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 , 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 pp 0=2λ/(2λ+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 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 $f\in H^{1}_{\lambda}({\mathbb{R}})$ .  相似文献   

10.
In this article we define and study the Dunkl convolution product and the Dunkl transform on spaces of distributions on By using the main results obtained, we study the hypoelliptic Dunkl convolution equations in the space of distributions.  相似文献   

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