Positivity of Dunkl’s intertwining operator via the trigonometric setting

نویسندگان

  • Margit Rösler
  • Michael Voit
چکیده

In this note, a new proof for the positivity of Dunkl’s intertwining operator in the crystallographic case is given. It is based on an asymptotic relationship between the Opdam-Cherednik kernel and the Dunkl kernel as recently observed by M. de Jeu, and on positivity results of S. Sahi for the Heckman-Opdam polynomials and their nonsymmetric counterparts. 2000 AMS Subject Classification: 33C52, 33C67.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Positivity of Dunkl’s Intertwining Operator

For a finite reflection group on R , the associated Dunkl operators are parametrized firstorder differential-difference operators which generalize the usual partial derivatives. They generate a commutative algebra which is – under weak assumptions – intertwined with the algebra of partial differential operators by a unique linear and homogeneous isomorphism on polynomials. In this paper it is s...

متن کامل

INTEGRATION OF THE INTERTWINING OPERATOR FOR h-HARMONIC POLYNOMIALS ASSOCIATED TO REFLECTION GROUPS

Let V be the intertwining operator with respect to the reflection invariant measure hαdω on the unit sphere S d−1 in Dunkl’s theory on spherical h-harmonics associated with reflection groups. Although a closed form of V is unknown in general, we prove that ∫ Sd−1 V f(y)hα(y)dω = Aα ∫ Bd f(x)(1 − |x|2)|α|1−1dx, where Bd is the unit ball of Rd and Aα is a constant. The result is used to show that...

متن کامل

Product Formula for Jacobi Polynomials, Spherical Harmonics and Generalized Bessel Function of Dihedral Type

Abstract. We work out the expression of the generalized Bessel function of B2-type derived in [4]. This is done using Dijskma and Koornwinder’s product formula for Jacobi polynomials and the obtained expression is given by multiple integrals involving only a normalized modified Bessel function and two symmetric Beta distributions. We think of that expression as the major step toward the explici...

متن کامل

Paley–wiener Theorems for the Dunkl Transform

We conjecture a geometrical form of the Paley–Wiener theorem for the Dunkl transform and prove three instances thereof, by using a reduction to the one-dimensional even case, shift operators, and a limit transition from Opdam’s results for the graded Hecke algebra, respectively. These Paley– Wiener theorems are used to extend Dunkl’s intertwining operator to arbitrary smooth functions. Furtherm...

متن کامل

Asymptotic Analysis for the Dunkl Kernel

This paper studies the asymptotic behavior of the integral kernel of the Dunkl transform, the so-called Dunkl kernel, when one of its arguments is fixed and the other tends to infinity either within a Weyl chamber of the associated reflection group, or within a suitable complex domain. The obtained results are based on the asymptotic analysis of an associated system of ordinary differential equ...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2004