The paper introduces new estimators for multivariate functions using Fourier methods.
problem Estimating multivariate functions like densities and regression functions.
method Monte Carlo estimators based on the Fourier integral theorem.
result Established rates of convergence for new estimators, often superior to existing methods.
The paper uses Fourier integral theorem for estimating multivariate distributions.
problem Estimating multivariate distributions and conditional distribution functions.
method Natural Monte Carlo and fully nonparametric estimators based on Fourier integral theorem.
result Explicit Monte Carlo estimators without estimated covariance matrix.
As announced in [12], we develop a calculus of Fourier integral G-operators on any Lie groupoid G. For that purpose, we study convolability and invertibility of Lagrangian conic submanifolds of the symplectic groupoid T * G. We also identify those Lagrangian which correspond to equivariant families parametrized by the …
Proof shows volume equals integral points for certain manifolds.
problem Counting integral points on affine manifolds.
method Rational Ehrhart theory and Fourier analysis.
result Volume equals number of integral points for integral-integral affine manifolds.
Transformers improve with Fourier integral attentions.
problem Inefficiency of dot-product attention in capturing feature dependencies.
method Interpreted attention as kernel regression, proposed FourierFormer with generalized Fourier integral kernels.
result FourierFormer achieves better accuracy and reduces redundancy.
New method for optimizing risk in financial models using Fourier transforms.
problem Optimizing risk in financial models with multi-period mean-CVaR.
method Strictly monotone 2D integration scheme via Fourier-trained transition kernels.
result Established robust and accurate optimization method for financial models.
We show that the index of an elliptic Fourier integral operator associated to a contact diffeomorphism φ of cosphere bundles of two Riemannian manifolds X and Y is given by ∫B∗XA^(T∗X)expθ−∫B∗YA^(T∗Y)expθ. Here B∗ stands for the unit coball bundle and θ is a certain characteristic…
Study evolution equations on Lie groupoids using Fourier integral operators.
problem Solving evolution equations on Lie groupoids.
method Developed calculus of Fourier integral operators and used them to study the fundamental solution of the evolution equation.
result Developed a method to find the fundamental solution of the evolution equation on Lie groupoids.
A new method integrates Fourier basis expansion and mapping for improved time series forecasting.
problem Inconsistent starting cycles and series length issues in Fourier-based methods.
method Fourier Basis Mapping (FBM) method that integrates time-frequency features through Fourier basis expansion and mapping.
result FBM addresses inconsistencies and preserves temporal characteristics, achieving SOTA performance.
X-ray transform on H-type groups solved, revealing function injectivity.
problem Injectivity in sub-Riemannian geometry.
method Fourier Slice Theorem adapted to H-type groups.
result Integrable functions on H-type groups are uniquely determined by their integrals over geodesics.
We endow the group of invertible Fourier integral operators on an open}manifold with the structure of an ILH Lie group. This is done by establishing such structures for the groups of invertible pseudodifferential operators and contact transformations on an open manifold of bounded geometry, and gluing those together vi…
Proves conditions for Fourier transforms in rank 1 symmetric spaces.
problem Understanding Fourier transform bounds in symmetric spaces.
method Proves sufficient and necessary conditions using Lipschitz and Fourier type integral conditions.
result Establishes bounds for Fourier transforms in rank 1 symmetric spaces with specific moduli of continuity.
New integral theorems improve density function estimations.
problem Improving density function estimations.
method Integrals based on cyclic functions and Riemann sums, Fourier integral theorem, Monte Carlo methods, variational approach, Cauchy residue theorem.
result Optimal cyclic functions minimize square integrals, improving density estimations.
Integrates Fourier features for faster Gaussian process regression.
problem Efficiently scaling Gaussian process regression to large datasets.
method Integrated Fourier features for Gaussian processes.
result Improves Gaussian process regression speed to O(M3) for a broad class of kernels. New proof of Zelditch's generalization using Riemannian geometry.
problem Asymptotic formula for eigenfunction sums on compact manifolds.
method Riemannian geometry methods.
result Proof of Zelditch's generalization without Fourier integral operators.
Study examines the geometry of a group of Fourier-integral operators related to Diff(S^1).
problem Investigating the geometry of a specific group of Fourier-integral operators.
method Defined a right-invariant pseudo-Riemannian metric on the group using renormalized traces of pseudo-differential operators.
result Extended the Hilbert-Schmidt Riemannian metric to the group.
EFiGP uses Fourier and eigen-decomposition for efficient ODE parameter estimation.
problem Parameter estimation and trajectory reconstruction for noisy, sparse, nonlinear ODE systems.
method EFiGP integrates Fourier transformation and eigen-decomposition into a physics-informed Gaussian Process framework.
result EFiGP efficiently estimates ODE parameters and recovers trajectories from noisy data.
The paper bounds Fourier integral operators on Hardy spaces with specific conditions.
problem Bounding Fourier integral operators on Hardy spaces with given conditions.
method Using Hörmander classes and phase conditions, the paper establishes boundedness of Fourier integral operators.
result The Fourier integral operator is bounded from local Hardy space hp to Lp under specified conditions. The square root of Fredholm determinants causes numerical instabilities in option pricing models.
problem Numerical instabilities in Fourier-based option pricing for the Volterra Stein-Stein model.
method Characterization of determinant crossing behavior, derivation of transform to handle crossings, efficient algorithms.
result Significant improvement in accuracy and reduction in computational cost for Fourier-based pricing.
Paper variates Navier-Stokes-Fourier system for thermodynamic consistency.
problem Modeling compressible fluid dynamics with thermodynamic constraints.
method Variational discretization with discrete exterior calculus.
result Derives a nonholonomic variational integrator for NSF system.
Study examines boundedness of oscillating singular integrals on specific Lie groups.
problem Investigating boundedness of oscillating singular integrals on Lie groups of polynomial growth.
method Presented kernel criteria in terms of sub-Riemannian structure and Fourier analysis.
result Extended classical oscillating conditions for boundedness of oscillating convolution operators.
Markov Chain Monte Carlo methods become increasingly popular in applied mathematics as a tool for numerical integration with respect to complex and high-dimensional distributions. However, application of MCMC methods to heavy tailed distributions and distributions with analytically intractable densities turns out to be…
Study analyzes Lévy process structure on manifolds with conjugate points.
problem Microlocal analysis of Lévy processes on manifolds with conjugate points.
method Microlocal analysis, pseudodifferential operators, Fourier integral operators.
result Generator can be expressed as sum of pseudodifferential and Fourier integral operators.
Quantum methods improve option pricing accuracy.
problem Pricing financial derivatives using Monte Carlo integration.
method Hybrid classical-quantum methods using Fourier series and QML.
result Quantum methods achieve remarkable accuracy in option pricing.
Study solves optimal portfolio selection using HJB equation.
problem Optimal portfolio selection problem.
method Maximal monotone operator method, Banach fixed-point theorem, Fourier transform, monotone operators technique.
result Existence and uniqueness of solution to HJB equation.
Computing accurate estimates of the Fourier transform of analog signals from discrete data points is important in many fields of science and engineering. The conventional approach of performing the discrete Fourier transform of the data implicitly assumes periodicity and bandlimitedness of the signal. In this paper, we…
A new Fourier model improves ODE prediction.
problem Improving the accuracy of ODE solutions, especially for periodic functions.
method Constructing a Fourier state space model and a hybrid model combining Taylor and Fourier methods.
result The hybrid model can predict ODE solutions more accurately, especially for periodic functions.
Study light ray transform in pseudo-Euclidean space, derive inversion formula, and prove stability.
problem Analyzing light ray transform in pseudo-Euclidean space.
method Investigate normal operator, derive inversion formula, analyze as Fourier Integral Operator.
result Derive an inversion formula and prove stability estimates.
FNN approximates functions and solves PDEs with periodic BCs.
problem Approximating and solving periodic functions and PDEs.
method Fourier neural network architecture with activation and loss functions.
result FNN can solve PDEs with periodic BCs and is interpretable.
Study extends neural network approximation to time-varying PDEs using Fourier-Lebesgue spaces.
problem Limitation to static PDEs and different time-domain regularity.
method Extend spectral Barron spaces to anisotropic weighted Fourier-Lebesgue spaces, measure approximation error in Bochner-Sobolev norm.
result Established bound on approximation rate for functions in anisotropic weighted Fourier-Lebesgue spaces.
Researchers extend asymptotic analysis to Bergman projections with Gevrey weights.
problem Analyzing Bergman projections with Gevrey weights.
method Extending direct approach to semiclassical asymptotics to Gevrey weights using Fourier integral operators.
result Gevrey symbol amplitude of asymptotic Bergman projection with Gevrey weights and Gevrey-type growth rate.
The paper introduces a new volatility model using Fourier techniques for pricing and hedging.
problem Pricing and hedging of financial derivatives with stochastic volatility.
method A Fourier-based approach to price and hedge European and path-dependent options in a stochastic volatility model.
result The model includes and extends popular volatility models like Stein-Stein, Bergomi, and Heston.
The paper studies Fourier-Laplace transforms in polynomial OU volatility models for option pricing.
problem Calibrating and pricing options in polynomial Ornstein-Uhlenbeck volatility models.
method Analyzes Fourier-Laplace transforms, connects to Riccati equations, and develops numerical schemes.
result Establishes existence and solution for Riccati equations and provides efficient numerical methods.
We provide an integral representation for the (implied) copulas of dependent random variables in terms of their moment generating functions. The proof uses ideas from Fourier methods for option pricing. This representation can be used for a large class of models from mathematical finance, including Lévy and affine proc…
The paper connects 3D manifold invariants to hyperbolic cone metrics and discrete Fourier transforms.
problem Volume conjecture for Reshetikhin-Turaev invariants of 3-manifolds with links.
method Volume conjecture, hyperbolic cone metrics, discrete Fourier transforms, change-of-pair operations.
result Volume conjecture proven for specific cases, provides approach to solving Volume Conjecture for hyperbolic 3-manifolds.
Study on estimating volatility of volatility using Fourier methods and provides insights into volatility dynamics.
problem Estimating the volatility of volatility (vol-of-vol) accurately and efficiently.
method Used Fourier methodology to estimate integrated volatility of volatility, bias-corrected and without bias-correction, comparing their asymptotic properties and accuracy.
result The bias-corrected estimator reaches the optimal rate n1/4, while the uncorrected estimator has a slower rate and smaller asymptotic variance. Study Fourier estimator for spot volatility with unbounded coefficients and jumps.
problem Estimating spot volatility with unbounded coefficients and jumps in price process.
method Fourier estimator for spot volatility, convergence analysis for unbounded coefficients and jumps.
result Convergence of trigonometric polynomial to volatility's path, almost sure convergence of reconstructed volatility.
We consider a compact Riemannian manifold with a Hermitian line bundle whose curvature is non-degenerate. The Laplacian acting on high tensor powers (the semiclassical regime) of the bundle exhibits a cluster of low-energy states. We demonstrate that the orthogonal projectors onto these states are the Fourier component…
Using equivariant Toeplitz operator calculus, we give a new proof of the Atiyah-Weinstein conjecture on the index of Fourier integral operators and the relative index of CR structures.
We give a detailed microlocal study of X-ray transforms over geodesics-like families of curves with conjugate points of fold type. We show that the normal operator is the sum of a pseudodifferential operator and a Fourier integral operator. We compute the principal symbol of both operators and the canonical relation as…
A new method solves complex financial equations efficiently.
problem Solving worst-case and best-case prices for two-factor uncertain volatility models.
method Decompose and integrate, then optimize; piecewise constant control; closed-form Green's functions; 2D convolution integrals; monotone numerical integration; Fast Fourier Transforms.
result The method efficiently computes the value function and optimal control, converging to the viscosity solution of the HJB equation.
We study the weighted light ray transform L of integrating functions on a Lorentzian manifold over lightlike geodesics. We analyze L as a Fourier Integral Operator and show that if there are no conjugate points, one can recover the spacelike singularities of a function f from its the weighted light ray transform …
This dissertation explores T-duality between hyperkähler structures and branes on algebraic integrable systems.
problem Investigates T-duality between hyperkähler structures and branes on algebraic integrable systems.
method Uses techniques of generalized geometry and Fourier-Mukai transform to show T-duality between semi-flat hyperkähler structures and generalized branes.
result Shows T-duality between semi-flat hyperkähler structures and generalized branes on algebraic integrable systems.
Developed a monotone numerical method for MV portfolio optimization under jump-diffusion models.
problem Efficiently optimizing portfolios with jump-diffusion dynamics and investment constraints.
method Strictly monotone numerical integration method using Fourier transforms and composite quadrature rules.
result Proven to be ℓ∞-stable and pointwise consistent, converging to the MV optimization solution. Paper proposes efficient method to calculate Fisher-Bingham distribution normalizing constant.
problem Efficiently calculating the normalizing constant of Fisher-Bingham distributions.
method Numerical integration with continuous Euler transform to Fourier-type integral representation.
result The method is fast and accurate, applicable to high-dimensional distributions.
New Fourier features improve high-precision approximation in large-scale problems.
problem Designing scalable, high-precision Fourier features for large-scale kernel methods.
method Introducing a new family of quadrature rules that accurately approximate the Gaussian measure in higher dimensions.
result Improved approximation bounds with new Fourier features.
We establish a rigorous link between infinite-dimensional regular Frölicher Lie groups built out of non-formal pseudodifferential operators and the Kadomtsev-Petviashvili hierarchy. We introduce a version of the Kadomtsev-Petviashvili hierarchy on a regular Frölicher Lie group of series of non-formal odd-class pseudodi…
A graphical calculus for microformal morphisms simplifies complex operations in classical and quantum physics.
problem Simplifying operations in classical and quantum microformal morphisms.
method Developed a graphical calculus inspired by Cattaneo-Dherin-Felder's work on formal symplectic groupoids, extended to quantum thick morphisms.
result Infinite series can be written as sums over bipartite trees for both classical and quantum thick morphisms.