Related papers: Spectral algorithms in higher-order Fourier analys…
We develop the basic building blocks of a frequency domain framework for drawing statistical inferences on the second-order structure of a stationary sequence of functional data. The key element in such a context is the spectral density…
In this note, we solve an inverse spectral problem for a class of finite band symmetric matrices. We provide necessary and sufficient conditions for a matrix valued function to be a spectral function of the operator corresponding to a…
Neural integral equations are deep learning models based on the theory of integral equations, where the model consists of an integral operator and the corresponding equation (of the second kind) which is learned through an optimization…
We address the task of higher-order derivative evaluation of computer programs that contain QR decompositions and real symmetric eigenvalue decompositions. The approach is a combination of univariate Taylor polynomial arithmetic and matrix…
We construct a functional model for rank one perturbations of compact normal operators acting in a certain Hilbert spaces of entire functions generalizing de Branges spaces. Using this model we study completeness and spectral synthesis…
The spectral decomposition for an explicit second-order differential operator $T$ is determined. The spectrum consists of a continuous part with multiplicity two, a continuous part with multiplicity one, and a finite discrete part with…
We consider a new class of Parareal algorithms, which use ideas from localized reduced basis methods to construct the coarse solver from spectral approximations of the transfer operators mapping initial values for a given time interval to…
The goal of this paper is to show that there exists a simple, yet universal statistical logic of spectral graph analysis by recasting it into a nonparametric function estimation problem. The prescribed viewpoint appears to be good enough to…
The aim of the paper is to investigate the solutions of special inhomogeneous linear functional equations by using spectral analysis in a translation invariant closed linear subspace of additive/multiadditive functions containing the…
Grover's quantum algorithm can find a marked item from an unstructured database faster than any classical algorithm, and hence it has been used for several applications such as cryptanalysis and optimization. When there exist multiple…
The problem of computing spectra of operators is arguably one of the most investigated areas of computational mathematics. However, the problem of computing spectra of general bounded infinite matrices has only recently been solved. We…
We introduce a notion of an algebra of generalized pseudo-differential operators and prove that a spectral triple is regular if and only if it admits an algebra of generalized pseudo-differential operators. We also provide a self-contained…
This article highlights some of the key operating principles of Grover algorithm. These principles were used to develop a new oracle function, that illustrates the possibility of using Grover algorithm for solving more realistic and…
Quantum information processing has been effectively demonstrated on a small number of qubits by nuclear magnetic resonance. An important subroutine in any computing is the readout of the output. ``Spectral implementation'' originally…
We describe how spectral functions of differential operators appear in the quantum field theory context. We formulate consistency conditions which should be satisfied by the operators and by the boundary conditions. We review some modern…
Differential equations may possess coefficients that vary on a spectrum of scales. Because coefficients are typically multiplicative in real space, they turn into convolution operators in spectral space, mixing all wavenumbers. However, in…
Exponential integrators based on contour integral representations lead to powerful numerical solvers for a variety of ODEs, PDEs, and other time-evolution equations. They are embarrassingly parallelizable and lead to global-in-time…
We propose a mathematical framework that we call quantum, higher-order Fourier analysis. This generalizes the classical theory of higher-order Fourier analysis, which led to many advances in number theory and combinatorics. We define a…
We introduce a formalism to produce several families of spectral sequences involving the derived functors of the limit and colimit functors over a finite partially ordered set. The first type of spectral sequences involves the left derived…
Over the decades, Functional Analysis has been enriched and inspired on account of demands from neighboring fields, within mathematics, harmonic analysis (wavelets and signal processing), numerical analysis (finite element methods,…