Sampling Theorem and Discrete Fourier Transform on the Riemann Sphere

Using coherent-state techniques, we prove a sampling theorem for Majorana's (holomorphic) functions on the Riemann sphere and we provide an exact reconstruction formula as a convolution product of $N$ samples and a given reconstruction kernel (a sinc-type function). We also discuss the effect of over- and under-sampling. Sample points are roots of unity, a fact which allows explicit inversion formulas for resolution and overlapping kernel operators through the theory of Circulant Matrices and Rectangular Fourier Matrices. The case of band-limited functions on the Riemann sphere, with spins up to $J$, is also considered. The connection with the standard Euler angle picture, in terms of spherical harmonics, is established through a discrete Bargmann transform.Comment: 26 latex pages. Final version published in J. Fourier Anal. App

Random matrix ensembles of time correlation matrices to analyze visual lifelogs

Visual lifelogging is the process of automatically recording images and other sensor data for the purpose of aiding memory recall. Such lifelogs are usually created using wearable cameras. Given the vast amount of images that are maintained in a visual lifelog, it is a significant challenge for users to deconstruct a sizeable collection of images into meaningful events. In this paper, random matrix theory (RMT) is applied to a cross-correlation matrix C, constructed using SenseCam lifelog data streams to identify such events. The analysis reveals a number of eigenvalues that deviate from the spectrum suggested by RMT. The components of the deviating eigenvectors are found to correspond to “distinct significant events” in the visual lifelogs. Finally, the cross-correlation matrix C is cleaned by separating the noisy part from the non-noisy part. Overall, the RMT technique is shown to be useful to detect major events in SenseCam images

Quantizing euclidean motions via double-coset decomposition

10.34133/2019/1608396Research2019160839

Black-scholes theory and diffusion processes on the cotangent bundle of the affine group

10.3390/E22040455Entropy22445