Related papers: The hyperanalytic signal
A particular mix of integral equations and discretization techniques is suggested for the solution of a planar Helmholtz transmission problem with relevance to the study of surface plasmon waves. The transmission problem describes the…
Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of…
For any orthogonal polynomials system on real line we construct an appropriate oscillator algebra such that the polynomials make up the eigenfunctions system of the oscillator hamiltonian. The general scheme is divided into two types: a…
We construct a class of Hamiltonians that describe the photodetection process from beginning to end. Our Hamiltonians describe the creation of a photon, how the photon travels to an absorber (such as a molecule), how the molecule absorbs…
Hypercomplex algebras have recently been gaining prominence in the field of deep learning owing to the advantages of their division algebras over real vector spaces and their superior results when dealing with multidimensional signals in…
In this paper, we propose a numerical method of Fourier transform based on hyperfunction theory. In the proposed method, we compute analytic functions called the defining functions, which give the desired Fourier transform as a…
Quaternion, an extension of complex number, is the first discovered non-commutative division algebra by William Rowan Hamilton in 1843. In this article, we review the recent progress on building up the connection between the mathematical…
We give a method of solution to the problem of iterating holomorphic functions to fractional or complex heights. We construct an auxiliary function from natural iterates of a holomorphic function; the auxiliary function will be…
One-dimensional signal decomposition is a well-established and widely used technique across various scientific fields. It serves as a highly valuable pre-processing step for data analysis. While traditional decomposition techniques often…
These notes present a first graduate course in harmonic analysis. The first part emphasizes Fourier series, since so many aspects of harmonic analysis arise already in that classical context. The Hilbert transform is treated on the circle,…
The Hilbert transform has been one of the foundational transforms in signal processing, finding it's way into multiple disciplines from cryptography to biomedical sciences. However, there does not exist any quantum analogue for the Hilbert…
Fourier phase retrieval is a classical problem that deals with the recovery of an image from the amplitude measurements of its Fourier coefficients. Conventional methods solve this problem via iterative (alternating) minimization by…
The ability to decompose a signal in an orthonormal basis (a set of orthogonal components, each normalized to have unit length) using a fast numerical procedure rests at the heart of many signal processing methods and applications. The…
An emerging way to deal with high-dimensional non-euclidean data is to assume that the underlying structure can be captured by a graph. Recently, ideas have begun to emerge related to the analysis of time-varying graph signals. This work…
A vector-valued signal in N dimensions is a signal whose value at any time instant is an N-dimensional vector, that is, an element of $\mathbb{R}^N$. The sum of an arbitrary number of such signals of the same frequency is shown to trace an…
We provide a description of phase transitions at finite temperature in strongly coupled field theories using holography. For this purpose, we introduce a general class of gravity duals to superconducting theories that exhibit various types…
Divergences are fundamental to the information criteria that underpin most signal processing algorithms. The alpha-beta family of divergences, designed for non-negative data, offers a versatile framework that parameterizes and continuously…
The changes in brightness of an astronomical source as a function of time are key probes into that source's physics. Periodic and quasi-periodic signals are indicators of fundamental time (and length) scales in the system, while stochastic…
The signal demixing problem seeks to separate a superposition of multiple signals into its constituent components. This paper studies a two-stage approach that first decompresses and subsequently deconvolves the noisy and undersampled…
In the framework of superanalysis we get a functions theory close to complex analysis, under a suitable condition (A) on the real superalgebras in consideration. Under the condition (A), we get an integral representation formula for the…