scholarly journals Dunkl-Sobolev spaces of exponential type and applications

2011 ◽  
Vol 9 (1) ◽  
pp. 41-66
Author(s):  
Hatem Mejjaoli
Keyword(s):  
Sobolev Spaces ◽  
Exponential Type ◽  
Reproducing Kernels ◽  
Difference Operators ◽  
Dunkl Operators

We study the Sobolev spaces of exponential type associated with the Dunkl operators. Some properties including completeness and imbedding theorem are proved. Next we introduce a classes of symbols of exponential type and the associated pseudo-differential-difference operators, which naturally act on the Dunkl-Sobolev spaces of exponential type. Finally using the theory of reproducing kernels some applications are given for these spaces.

Weighted higher order exponential type inequalities in metric spaces and applications

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Huiju Wang ◽  
Pengcheng Niu
Keyword(s):  
Homogeneous Space ◽  
Lie Groups ◽  
Sobolev Spaces ◽  
Exponential Type ◽  
Metric Spaces ◽  
Type Inequality ◽  
Higher Order ◽  
Geodesic Space ◽  
Stratified Lie Groups ◽  
Weighted Case

AbstractIn this paper, we establish weighted higher order exponential type inequalities in the geodesic space {({X,d,\mu})} by proposing an abstract higher order Poincaré inequality. These are also new in the non-weighted case. As applications, we obtain a weighted Trudinger’s theorem in the geodesic setting and weighted higher order exponential type estimates for functions in Folland–Stein type Sobolev spaces defined on stratified Lie groups. A higher order exponential type inequality in a connected homogeneous space is also given.

scholarly journals Reproducing kernels of Sobolev spaces on ℝd and applications to embedding constants and tractability

2018 ◽  
Vol 16 (05) ◽  
pp. 693-715 ◽  
Author(s):  
Erich Novak ◽  
Mario Ullrich ◽  
Henryk Woźniakowski ◽  
Shun Zhang
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Reproducing Kernel ◽  
Absolute Error ◽  
Reproducing Kernels ◽  
Worst Case ◽  
Reproducing Kernel Formula ◽  
Positive Integers ◽  
Integration Errors ◽  
Exponentially Small

The standard Sobolev space [Formula: see text], with arbitrary positive integers [Formula: see text] and [Formula: see text] for which [Formula: see text], has the reproducing kernel [Formula: see text] for all [Formula: see text], where [Formula: see text] are components of [Formula: see text]-variate [Formula: see text], and [Formula: see text] with non-negative integers [Formula: see text]. We obtain a more explicit form for the reproducing kernel [Formula: see text] and find a closed form for the kernel [Formula: see text]. Knowing the form of [Formula: see text], we present applications on the best embedding constants between the Sobolev space [Formula: see text] and [Formula: see text], and on strong polynomial tractability of integration with an arbitrary probability density. We prove that the best embedding constants are exponentially small in [Formula: see text], whereas worst case integration errors of algorithms using [Formula: see text] function values are also exponentially small in [Formula: see text] and decay at least like [Formula: see text]. This yields strong polynomial tractability in the worst case setting for the absolute error criterion.

scholarly journals Laguerre semigroup and Dunkl operators

2012 ◽  
Vol 148 (4) ◽  
pp. 1265-1336 ◽  
Author(s):  
Salem Ben Saïd ◽  
Toshiyuki Kobayashi ◽  
Bent Ørsted
Keyword(s):  
Unitary Representation ◽  
Fourier Transforms ◽  
Universal Covering ◽  
Hankel Transform ◽  
Kernel Functions ◽  
Difference Operators ◽  
Dunkl Transform ◽  
Dunkl Operators ◽  
Hermite Semigroup ◽  
Heisenberg Uncertainty

AbstractWe construct a two-parameter family of actionsωk,aof the Lie algebra 𝔰𝔩(2,ℝ) by differential–difference operators on ℝN∖{0}. Herekis a multiplicity function for the Dunkl operators, anda>0 arises from the interpolation of the two 𝔰𝔩(2,ℝ) actions on the Weil representation ofMp(N,ℝ) and the minimal unitary representation of O(N+1,2). We prove that this actionωk,alifts to a unitary representation of the universal covering ofSL(2,ℝ) , and can even be extended to a holomorphic semigroup Ωk,a. In thek≡0 case, our semigroup generalizes the Hermite semigroup studied by R. Howe (a=2) and the Laguerre semigroup studied by the second author with G. Mano (a=1) . One boundary value of our semigroup Ωk,aprovides us with (k,a) -generalized Fourier transforms ℱk,a, which include the Dunkl transform 𝒟k(a=2) and a new unitary operator ℋk (a=1) , namely a Dunkl–Hankel transform. We establish the inversion formula, a generalization of the Plancherel theorem, the Hecke identity, the Bochner identity, and a Heisenberg uncertainty relation for ℱk,a. We also find kernel functions for Ωk,aand ℱk,afora=1,2 in terms of Bessel functions and the Dunkl intertwining operator.

scholarly journals Reproducing kernel Hilbert spaces on manifolds: Sobolev and diffusion spaces

2020 ◽  
pp. 1-34
Author(s):  
Ernesto De Vito ◽  
Nicole Mücke ◽  
Lorenzo Rosasco
Keyword(s):  
Sobolev Spaces ◽  
Riemannian Manifolds ◽  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Differential Operators ◽  
Reproducing Kernels ◽  
Reproducing Kernel Hilbert Spaces ◽  
Kernel Hilbert Spaces ◽  
Euclidean Setting ◽  
And Diffusion

We study reproducing kernel Hilbert spaces (RKHS) on a Riemannian manifold. In particular, we discuss under which condition Sobolev spaces are RKHS and characterize their reproducing kernels. Further, we introduce and discuss a class of smoother RKHS that we call diffusion spaces. We illustrate the general results with a number of detailed examples. While connections between Sobolev spaces, differential operators and RKHS are well known in the Euclidean setting, here we present a self-contained study of analogous connections for Riemannian manifolds. By collecting a number of results in unified a way, we think our study can be useful for researchers interested in the topic.

FRACTIONAL FOCK–SOBOLEV SPACES

2018 ◽  
Vol 237 ◽  
pp. 79-97
Author(s):  
HONG RAE CHO ◽  
SOOHYUN PARK
Keyword(s):  
Sobolev Spaces ◽  
Reproducing Kernels ◽  
Fock Spaces ◽  
Size Estimate ◽  
Weighted Fock Spaces

Let $s\in \mathbb{R}$ and $0<p\leqslant \infty$. The fractional Fock–Sobolev spaces $F_{\mathscr{R}}^{s,p}$ are introduced through the fractional radial derivatives $\mathscr{R}^{s/2}$. We describe explicitly the reproducing kernels for the fractional Fock–Sobolev spaces $F_{\mathscr{R}}^{s,2}$ and then get the pointwise size estimate of the reproducing kernels. By using the estimate, we prove that the fractional Fock–Sobolev spaces $F_{\mathscr{R}}^{s,p}$ are identified with the weighted Fock spaces $F_{s}^{p}$ that do not involve derivatives. So, the study on the Fock–Sobolev spaces is reduced to that on the weighted Fock spaces.

Sign in / Sign up

Export Citation Format

Share Document