Clifford algebras, Fourier transforms and quantum mechanics

In this review, an overview is given of several recent generalizations of the Fourier transform, related to either the Lie algebra sl_2 or the Lie superalgebra osp(1|2). In the former case, one obtains scalar generalizations of the Fourier transform, including the fractional Fourier transform, the Dunkl transform, the radially deformed Fourier transform and the super Fourier transform. In the latter case, one has to use the framework of Clifford analysis and arrives at the Clifford-Fourier transform and the radially deformed hypercomplex Fourier transform. A detailed exposition of all these transforms is given, with emphasis on aspects such as eigenfunctions and spectrum of the transform, characterization of the integral kernel and connection with various special functions.Comment: Review paper, 39 pages, to appear in Math. Methods. Appl. Sc

Coded Fourier Transform

We consider the problem of computing the Fourier transform of high-dimensional vectors, distributedly over a cluster of machines consisting of a master node and multiple worker nodes, where the worker nodes can only store and process a fraction of the inputs. We show that by exploiting the algebraic structure of the Fourier transform operation and leveraging concepts from coding theory, one can efficiently deal with the straggler effects. In particular, we propose a computation strategy, named as coded FFT, which achieves the optimal recovery threshold, defined as the minimum number of workers that the master node needs to wait for in order to compute the output. This is the first code that achieves the optimum robustness in terms of tolerating stragglers or failures for computing Fourier transforms. Furthermore, the reconstruction process for coded FFT can be mapped to MDS decoding, which can be solved efficiently. Moreover, we extend coded FFT to settings including computing general $n$-dimensional Fourier transforms, and provide the optimal computing strategy for those settings

Steerable Discrete Fourier Transform

Directional transforms have recently raised a lot of interest thanks to their numerous applications in signal compression and analysis. In this letter, we introduce a generalization of the discrete Fourier transform, called steerable DFT (SDFT). Since the DFT is used in numerous fields, it may be of interest in a wide range of applications. Moreover, we also show that the SDFT is highly related to other well-known transforms, such as the Fourier sine and cosine transforms and the Hilbert transforms

Compressive Fourier Transform Spectroscopy

We describe an approach based on compressive-sampling which allows for a considerable reduction in the acquisition time in Fourier-transform spectroscopy. In this approach, an N-point Fourier spectrum is resolved from much less than N time-domain measurements using a compressive-sensing reconstruction algorithm. We demonstrate the technique by resolving sparse vibrational spectra using <25% of the Nyquist rate samples in single-pulse CARS experiments. The method requires no modifications to the experimental setup and can be directly applied to any Fourier-transform spectroscopy measurement, in particular multidimensional spectroscopy

The cylindrical Fourier transform

In this paper we devise a so-called cylindrical Fourier transform within the Clifford analysis context. The idea is the following: for a fixed vector in the image space the level surfaces of the traditional Fourier kernel are planes perpendicular to that fixed vector. For this Fourier kernel we now substitute a new Clifford-Fourier kernel such that, again for a fixed vector in the image space, its phase is constant on co-axial cylinders w.r.t. that fixed vector. The point is that when restricting to dimension two this new cylindrical Fourier transform coincides with the earlier introduced Clifford-Fourier transform.We are now faced with the following situation: in dimension greater than two we have a first Clifford-Fourier transform with elegant properties but no kernel in closed form, and a second cylindrical one with a kernel in closed form but more complicated calculation formulae. In dimension two both transforms coincide. The paper concludes with the calculation of the cylindrical Fourier spectrum of an L2-basis consisting of generalized Clifford-Hermite functions

Holomorphic harmonic analysis on complex reductive groups

We define the holomorphic Fourier transform of holomorphic functions on complex reductive groups, prove some properties like the Fourier inversion formula, and give some applications. The definition of the holomorphic Fourier transform makes use of the notion of $K$-admissible measures. We prove that $K$-admissible measures are abundant, and the definition of holomorphic Fourier transform is independent of the choice of $K$-admissible measures.Comment: 15 pages, revision of a preprint by the first author in 200