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The Whittaker Gabor transform (WGT) is a novel addition to the class of Gabor transforms, which has gained a respectable status in the realm of time-frequency signal analysis within a short span of time. Knowing the fact that the study of the time-frequency analysis is both theoretically interesting and practically useful, the aim of this article is to explore two more aspects of the time-frequency analysis associated with the WGT including the spectral analysis associated with the concentration operators and the scalogram. [ABSTRACT FROM AUTHOR]

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]

We study Bessel processes on Weyl chambers of types A and B on R N . Using elementary symmetric functions, we present several space-timeharmonic functions and thus martingales for these processes (X t) t ≥ 0 which are independent from one parameter of these processes. As a consequence, p t (y) : = E (∏ i = 1 N (y - X t i)) can be expressed via classical orthogonal polynomials. Such formulas on characteristic polynomials admit interpretations in random matrix theory where they are partially known by Diaconis, Forrester, and Gamburd. [ABSTRACT FROM AUTHOR]

Multivariate Bessel processes (X t , k) t ≥ 0 describe interacting particle systems of Calogero-Moser-Sutherland type and are related with β-Hermite and β-Laguerre ensembles. They depend on a root system and a multiplicity k. Recently, several limit theorems were derived for k → ∞ with fixed starting point. Moreover, the SDEs of (X t , k) t ≥ 0 were used to derive strong laws of large numbers for k → ∞ with starting points of the form k · x with x in the interior of the Weyl chambers. Here we provide associated almost sure functional central limit theorems which are locally uniform in t. The Gaussian limit processes admit explicit representations in terms of the solutions of associated deterministic ODEs. [ABSTRACT FROM AUTHOR]

Localization operators have a relatively recent development in pure and applied mathematics. Motivated by Wong's approach, we will study in this paper the time–frequency analysis associated with the hypergeometric Wigner transform related to the Cherednik operators in the case of the root system B C d . [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of the Dunkl weight function K c , λ for any irreducible representation λ of any finite Coxeter group W, generalizing previous results of Dunkl. In particular, K c , λ is a family of tempered distributions on the real reflection representation of W taking values in End C (λ) , with holomorphic dependence on the complex multi-parameter c. When the parameter c is real, the distribution K c , λ provides an integral formula for Cherednik's Gaussian inner product γ c , λ on the Verma module Δ c (λ) for the rational Cherednik algebra H c (W , h) .queryPlease check and confirm the inserted city name 'Stanford' for the affiliation is correct. In this case, the restriction of K c , λ to the hyperplane arrangement complement h R , r e g is given by integration against an analytic function whose values can be interpreted as braid group invariant Hermitian forms on K Z (Δ c (λ)) , where KZ denotes the Knizhnik–Zamolodchikov functor introduced by Ginzburg–Guay–Opdam–Rouquier. This provides a concrete connection between invariant Hermitian forms on representations of rational Cherednik algebras and invariant Hermitian forms on representations of Iwahori–Hecke algebras, and we exploit this connection to show that the KZ functor preserves signatures, and in particular unitarizability, in an appropriate sense. [ABSTRACT FROM AUTHOR]

Using the harmonic analysis associated to the Cherednik operators in the W-invariant case, relating to the root system, we define and study the Wigner and Weyl transformswhereσis a symbol in. We give the connection between these transforms, and criterias in terms of the symbolσto prove the boundedness and compactness of the transform. [ABSTRACT FROM PUBLISHER]

We present an explicit difference equation for the Heckman–Opdam hypergeometric function associated with root systems. Via a confluent hypergeometric limit, an analogous difference equation is obtained for the class-one Whittaker function diagonalizing the open quantum Toda chain. [ABSTRACT FROM AUTHOR]

This paper is perhaps the first attempt at a study of the Hardy space $$H^1$$ in the rational Dunkl setting. Following Uchiyama's approach, we characterize $$H^1$$ atomically and by means of the heat maximal operator. We also obtain a Fourier multiplier theorem for $$H^1$$ . These results are proved here in the one-dimensional case and in the product case. [ABSTRACT FROM AUTHOR]

We consider the transmutation operators Vk and t Vk associated with the Cherednik operators attached to the root system A2, called also in [Adv. Pure Appl. Math. 1 (2010), 293-323; Adv. Pure Appl. Math. 2 (2011), 223-246] the trigonometric Dunkl intertwining operator, and its dual. In this paper, we prove that the operators Vk and t Vk are positivity preserving and allows positive integral representations. In particular, we deduce that the Opdam-Cherednik kernel is positive definite. [ABSTRACT FROM AUTHOR]

Through the theory of Jack polynomials we give an iterative method for integral formula of Bessel function of typeAN−1and a partial product formula for it. [ABSTRACT FROM AUTHOR]

Using a generalized spherical mean operator, we define generalized modu-lus of smoothness in the space $\mathrm {L}_{k}^{2}(\mathbb {R}^{d})$. Based on the Dunkl operator we define Sobolev-type space and K-functionals. The main result of the paper is the proof of the equivalence theorem for a K-functional and a modulus of smoothness for the Dunkl transform on ℝ. [ABSTRACT FROM AUTHOR]

We study the canonical U-valued differential form, whose projections to different Kac-Moody algebras are key ingredients of the hypergeometric integral solutions of KZ-type differential equations and Bethe ansatz constructions. We explicitly determine the coefficients of the projections in the simple Lie algebrasA r,B r,C r,D r in a conveniently chosen Poincaré- Birchoff-Witt basis. [ABSTRACT FROM AUTHOR]