In this paper we prove a new inversion theorem and a refinement of an old support theorem for two Radon transforms on a symmetric space. Included are some new identities for the Abel transform and some results about the Fourier transform from a joint work with Rawat, Sengupta and Sitaram.
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48 results for “inverse Fourier transform”
Structured CNN designed using the prior information of problems potentially improves efficiency over conventional CNNs in various tasks in solving PDEs and inverse problems in signal processing. This paper introduces BNet2, a simplified Butterfly-Net and inline with the conventional CNN. Moreover, a Fourier transform i…
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.
Efficient inverse -transform and pricing barrier and lookback options with discrete monitoringmath.NA
The paper derives formulas for option pricing and random walk expectations.
problem Calculating the price of barrier and lookback options.
method Inverse Z-transform, Fourier/Laplace inversion, Wiener-Hopf factorization, and numerical methods.
result Efficient numerical methods for option pricing are developed.
Study spherical Fourier transform on hypergeometric type harmonic manifolds.
problem Spherical Fourier transform on harmonic Hadamard manifolds.
method Representation of spherical functions by Gauss hypergeometric functions.
result Inversion formula, convolution rule, and Plancherel theorem are derived.
Analytical pricing formulas and Greeks are obtained for European and American basket put options using Mellin transforms. We assume assets are driven by geometric Brownian motion which exhibit correlation and pay a continuous dividend rate. A novel approach to numerical Mellin inversion is achieved via the fast Fourier…
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 consider the horospherical transform and its inversion in 3 examples of hyperboloids. We want to illustrate via these examples the fact that the horospherical inversion formulas can be directly extracted from the classical Radon inversion formula. In a more broad context, this possibility reflects the fact that the …
FourNet approximates financial transition densities using Fourier transforms.
problem Approximating transition densities in finance with high accuracy.
method FourNet is a novel FFNN with Gaussian activation, learning from characteristic functions.
result FourNet can approximate transition densities arbitrarily well with finite neurons.
We extend the Fourier cosine method to discrete probability distributions, achieving faster convergence rates.
problem Extending Fourier cosine method to discrete probability distributions.
method Spectral filters and convergence rates analysis.
result Spectral filters achieve one order faster convergence rates than previously recognized.
Solves wave equation on non-flat harmonic manifolds using Abel transform and Fourier analysis.
problem Wave equation on non-flat harmonic manifolds with specific curvature conditions.
method Explicit representation using inverse dual Abel transform and Fourier transform.
result Shows asymptotic Huygens principle and equidistribution of energy.
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.
A unified Fourier slice method to derive ridgelet transform for a variety of depth-2 neural networkscs.LG
Unified method for deriving ridgelet transforms for various neural network architectures.
problem Deriving closed-form expressions for ridgelet transforms in modern neural network architectures.
method Unified Fourier slice method to derive ridgelet transforms for diverse neural network types.
result Systematic method to derive ridgelet transforms for various neural network architectures.
We establish several closed pricing formula for various path-independent payoffs, under an exponential Lévy model driven by the Variance Gamma process. These formulas take the form of quickly convergent series and are obtained via tools from Mellin transform theory as well as from multidimensional complex analysis. Par…
We provide a bound for the error committed when using a Fourier method to price European options when the underlying follows an exponential \levy dynamic. The price of the option is described by a partial integro-differential equation (PIDE). Applying a Fourier transformation to the PIDE yields an ordinary differential…
Previous research has shown that computation of convolution in the frequency domain provides a significant speedup versus traditional convolution network implementations. However, this performance increase comes at the expense of repeatedly computing the transform and its inverse in order to apply other network operati…
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.
A new class of harmonic Hadamard manifolds, those spaces called of hypergeometric type, is defined in terms of Gauss hypergeometric equations. Spherical Fourier transform defined on a harmonic Hadamard manifold of hypergeometric type admits an inversion formula. A characterization of harmonic Hadamard manifold being of…
HFNO enhances interpretability of turbulent flows through parallel wavenumber bin processing.
problem Opaque inner workings of Fourier Neural Operators (FNOs) hinder physical interpretability.
method Introduces HFNO, a novel FNO-based architecture that processes wavenumber bins in parallel, enhancing interpretability.
result HFNO decomposes turbulent flows across various scales, enabling increased interpretability and multiscale modeling.
Paper proves Fourier transform for valuations, simplifying previous work.
problem Existence of isomorphism for translation-invariant smooth valuations.
method Directly describes Alesker's isomorphism in terms of Fourier transform on functions.
result Simple proofs of Alesker's Fourier transform properties, including a previously conjectured result.
Initialization of parameters in deep neural networks has been shown to have a big impact on the performance of the networks (Mishkin & Matas, 2015). The initialization scheme devised by He et al, allowed convolution activations to carry a constrained mean which allowed deep networks to be trained effectively (He et al.…
Novel neural network layer improves long-range interactions in point clouds.
problem Efficiently treating long-range interactions in point clouds.
method Long-range convolutional (LRC)-layer that incorporates Fourier transforms.
result Global all-to-all convolution operation performed in nearly-linear time.
iSTFTNet speeds up mel-spectrogram vocoders without sacrificing quality.
problem Efficiently converting mel-spectrograms to speech with minimal computation.
method Replaces convolutional layers with iSTFT after frequency dimension reduction.
result Significant reduction in computational cost with comparable quality.
This paper explores deep learning for improving X-ray CT image reconstruction from undersampled data.
problem Improving image reconstruction from undersampled X-ray CT data.
method Analysis of classical and deep learning methods for solving inverse problems.
result Deep learning methods show promise in improving image quality from undersampled data.
Established PFPPs in complete markets, solving integral equations.
problem Existence of Predictable Forward Performance Processes in complete markets.
method Solving a one-period integral equation using Fourier transform for tempered distributions.
result Closed-form solutions for PFPPs with inverse marginal functions that are completely monotonic.
A new algorithm computes Fourier coefficients for a specified range efficiently.
problem Inefficiency in FFT due to fixed output size for all applications.
method Fast Partial Fourier Transform (PFT) that allows specifying the range of Fourier coefficients to compute.
result PFT achieves significant speedup over state-of-the-art FFT algorithms for small output sizes.
Paper develops methods for estimating and simulating a Student-t Lévy regression model.
problem Estimation and simulation of Student-t Lévy process with arbitrary degrees of freedom.
method Develops a two-step estimation procedure and simulates increments using inverse Fourier transform.
result Efficient estimation and simulation methods for Student-t Lévy process.
We study the microlocal properties of the geodesic X-ray transform on a manifold with boundary allowing the presence of conjugate points. Assuming that there are no self-intersecting geodesics and all conjugate pairs are nonsingular we show that the normal operator $\mathcal{N} = \mathcal{X}^t \circ \math…
Paper computes link determinants using Fourier-Hadamard transforms.
problem Computing determinants of complex link structures.
method Fourier-Hadamard transforms of Boolean functions.
result Determinant of centrally symmetric links with even components equals zero.
Develops flexible non-parametric ACFs using B-spline kernels.
problem Flexible modelling of the autocovariance function (ACF) in time-series, spatial, and spatio-temporal analysis.
method Derives the inverse Fourier transform of B-spline spectral bases to create a general class of non-parametric ACFs.
result Provides a provably dense, flexible, and general class of non-parametric ACFs for various types of processes.
The Fourier transform of Heegaard Floer d-invariants helps classify 3-manifolds.
problem Classifying 3-manifolds up to integer homology cobordism.
method Exploring the Fourier transform of Heegaard Floer d-invariants.
result Lens spaces are cancellable in the monoid of 3-manifolds up to integer homology cobordism.
The sub-Riemannian X-ray Transform on H-type groups: Fourier-Slice Theorems and Injectivity setsmath.DG
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.
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.
Quasi-Monte Carlo with Domain Transformation for Efficient Fourier Pricing of Multi-Asset Optionsq-fin.CP
This work improves Fourier pricing for multi-asset options using RQMC with domain transformation.
problem Efficiently pricing multi-asset options in high dimensions with Fourier methods.
method Randomized quasi-Monte Carlo (RQMC) with domain transformation to handle singularities.
result RQMC with domain transformation provides accurate and scalable Fourier pricing for multi-asset options.
Algorithm describes Fourier transform of Stokes data at infinity.
problem Understanding the Fourier transform of Stokes data at infinity.
method Topological description and algorithmic approach using recent results and language of Stokes local systems.
result Explicit isomorphisms between wild character varieties are induced.
We characterize stationary solutions to McKean-Vlasov equations on the circle.
problem Stationary solutions of McKean-Vlasov equations on the circle.
method Exact equivalence to an infinite-dimensional quadratic system of equations over Fourier coefficients, leading to explicit characterization of stationary states.
result Analytic expressions for the emergence, form, and shape of bifurcations involving multiple Fourier modes, and connections with discontinuous phase transitions.
New algorithms learn sparse set functions in non-orthogonal Fourier bases.
problem Learning sparse set functions in non-orthogonal Fourier bases.
method Novel algorithms using non-orthogonal Fourier transforms.
result At most queries for non-zero Fourier coefficients.
In this paper, we study robust tensor completion by using transformed tensor singular value decomposition (SVD), which employs unitary transform matrices instead of discrete Fourier transform matrix that is used in the traditional tensor SVD. The main motivation is that a lower tubal rank tensor can be obtained by usin…
The Plancherel theorem for Fourier--Laplace--Nahm transform for connections on the projective linemath.DG
We prove that the Fourier--Laplace--Nahm transform for connections on the projective line is a hyper-Kähler isometry.
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.
The paper derives statistics of multi-factor functions from their Fourier transforms.
problem Deriving statistics of multi-factor functions from Fourier transforms.
method Developed an m-Coefficient/Index Annihilation Theorem to analyze the moments of a function from its Fourier transform.
result The mth moment of a function becomes a series of terms, each with precisely m Fourier coefficients, and the indices sum to zero.
Efficient trainable front-end for neural speech enhancement.
problem Inefficient STFT front-ends in neural speech enhancement models.
method Butterfly mechanism for Fast Fourier Transform, trainable STFT window.
result Accuracy and efficiency improvements for low-compute systems.
Given two compact hyperkähler surfaces and and a holomorphic vector bundle on , which is a generalized instanton, one can define a Fourier-Mukai transform, which, under suitable assumptions, maps vector bundles on to vector bundles on . If and are dual complex tori, this transform …
Steganography is the science of hiding a secret message within an ordinary public message, which is referred to as Carrier. Traditionally, digital signal processing techniques, such as least significant bit encoding, were used for hiding messages. In this paper, we explore the use of deep neural networks as steganograp…
Graph Signal Processing improves stock market volatility forecasting.
problem Forecasting realized volatility in a global stock market context.
method Integrating Graph Signal Processing into the HAR model.
result The proposed model outperforms HAR-type benchmarks.
Enhances Fourier estimator performance for asynchronous event-data.
problem Improving correlation and covariance estimation on event-data.
method Implement and test NUFFT methods with different averaging kernels.
result Demonstrates improved performance and relationship between averaging scales.
Many signals on Cartesian product graphs appear in the real world, such as digital images, sensor observation time series, and movie ratings on Netflix. These signals are "multi-dimensional" and have directional characteristics along each factor graph. However, the existing graph Fourier transform does not distinguish …
RP-GFRFT unifies fractional order and rotation control for graph signals.
problem Lack of rotation-based spectral control in GFRFT and zero-angle degeneracy in AGFT.
method Rotation-parameterized graph fractional Fourier transform (RP-GFRFT) with degeneracy preserving rotation matrix.
result RP-GFRFT improves spectral filtering performance over existing methods.