Generalized Fourier Transforms Arising from the Enveloping Algebras of 𝔰𝔩(2) and 𝔬𝔰𝔭(1∣2)

Abstract

The Howe dual pair (sl (2), O(m)) allows the characterization of the classical Fourier transform (FT) on the space of rapidly decreasing functions as the exponential of a wellchosen element of sl(2) such that the Helmholtz relations are satisfied.In this paper we first investigate what happens when instead we consider exponentials of elements of the universal enveloping algebra of sl(2). This leads to a complete class of generalized Fourier transforms, that all satisfy properties similar to the classical FT. There is moreover a finite subset of transforms which very closely resemble the FT. We obtain operator exponential expressions for all these transforms by making extensive use of the theory of integer-valued polynomials. We also find a plane wave decomposition of their integral kernel and establish uncertainty principles. In important special cases we even obtain closed formulas for the integral kernels.In the second part of the paper, the same problem is considered for the dual pair (osp(1|2), Spin(m)), in the context of the Dirac operator on R m . This connects our results with the Clifford-Fourier transform studied in previous work.with ∆ x the Laplace operator and E x the Euler operator, one haswhich relates the transform with the representation theory of the Lie algebra sl(2) as the oper-[23, 28] for a detailed mathematical treatment. By now well-established fields of research based on this observation include the fractional FT [39] and the family of linear canonical transforms (LCTs) [45]. Both have extensive practical applications in the design of optical systems and signal processing.The observation in formula (1.1) has also lead to many other theoretically oriented generalizations of the FT, by considering alternative realizations of sl(2) in terms of differential or difference operators (or combinations thereof). Once such a realization is obtained, it is possible to define a generalized FT based on formula (1.1). It is then a challenging question to find a concrete integral transform expression for this operator. Research on this topic has a long history: it has been investigated in both the continuous and the discrete case, and also for more complicated algebras than sl (2), such as the superalgebra osp(1|2). Important examples in the continuous case include the Dunkl transform [19,2] and the radially deformed Fourier transform [35,36,3,14] as well as its Clifford deformation [16,17]. In more complicated geometries, the super Fourier transform [13] and a q-deformed version [10] have been investigated. In the discrete case we mention the fractional Fourier-Kravchuk transform [1] and various further deformations [32,33]. For a more detailed review of the followed strategy and the results in this line of investigation we refer the reader to [37,12].The present paper has a related yet different aim. We do not intend to change the realization of sl(2) in formula (1.2), but instead wish to study precisely how unique the operator realization of the FT in (1.1) is and to what extent the interplay of sl(2)...

Clifford–Fourier transform on hyperbolic space

“…The new kernel is still an eigenfunction of the hyperbolic Laplacian, The Plancherel measure is still the same because the Gamma operator commutes with radial functions. Equally, in the vein of , we can define with P ( x ) is a polynomial. We can choose P ( x ) such that P (Γ) is the Casimir operator of the S p i n ( m ) representation.…”

“…The Plancherel measure is still the same because the Gamma operator commutes with radial functions. Equally, in the vein of [14], we can define…”

Uncertainty Principles for the Clifford–Fourier Transform

“…In the even dimension case, they give an inversion formula for the Clifford-Fourier transform. Heisenberg's inequality [6,10], Donoho and Stark's theorem and Benedicks's theorem [13] are obtained for the Clifford-Fourier transform. The purpose of this paper is to generalize Hardy's theorem and Miyachi's theorem for the Clifford-Fourier transform on R m .…”

A new construction of the Clifford-Fourier kernel

“…In later work, the results were extended to fractional versions of the Clifford-Fourier transform [5,7] and integral kernels satisfying certain generalized Helmholtz PDEs in Clifford analysis [6]. In [8] an approach using Lie superalgebras and group symmetries led to a complete classification of transforms that behave in the same way as both the Clifford-Fourier transform and the classical Fourier transform. Uncertainty principles for these transforms were obtained in [12,8].…”