Related papers: A double Fourier sphere method for $d$-dimensional…
We investigate analytic properties of the double Fourier sphere (DFS) method, which transforms a function defined on the two-dimensional sphere to a function defined on the two-dimensional torus. Then the resulting function can be written…
We analyze the Double Fourier Sphere (DFS) method on the rotation group $\mathcal{SO}(3)$ in the frequency domain and demonstrate its central role in fast algorithms. Fast Fourier algorithms on $\mathcal{SO}(3)$ are commonly formulated as a…
Calculations of the Fourier transform of a constant quantity over an area or volume defined by polygons (connected vertices) are often useful in modeling wave scattering, or in fourier-space filtering of real-space vector-based volumes and…
Fourier series multiscale method, a concise and efficient analytical approach for multiscale computation, will be developed out of this series of papers. The second paper is concerned with simultaneous approximation to functions and their…
Fourier transform (FT) plays a crucial role in a broad range of applications, from enhancement, restoration and analysis through to security, compression and manipulation. The Fourier transform (FT) is a process that converts a function…
A collection of algorithms is described for numerically computing with smooth functions defined on the unit sphere. Functions are approximated to essentially machine precision by using a structure-preserving iterative variant of Gaussian…
The well-known discrete Fourier transform (DFT) can easily be generalized to arbitrary nodes in the spatial domain. The fast procedure for this generalization is referred to as nonequispaced fast Fourier transform (NFFT). Various…
In many applications data are measured or defined on a spherical manifold; spherical harmonic transforms are then required to access the frequency content of the data. We derive algorithms to perform forward and inverse spin spherical…
The work develops further the theory of the following inversion problem, which plays the central role in the rapidly developing area of thermoacoustic tomography and has intimate connections with PDEs and integral geometry: {\it Reconstruct…
This paper presents a "two-dimensional Fourier Continuation" method (2D-FC) for construction of bi-periodic extensions of smooth non-periodic functions defined over general two-dimensional smooth domains. The approach can be directly…
We propose a patchwise local Fourier extension method for approximating smooth functions on general two dimensional domains with curved boundaries. The domain is embedded into a Cartesian background grid and decomposed into rectangular…
Quasi-periodic trajectories with two or more incommensurate frequencies are ubiquitous in nonlinear dynamics, yet the classical Fourier-based time-spectral method is tied to strictly periodic responses. We introduce a torus time-spectral…
We provide a rigorous convergence proof demonstrating that the well-known semi-analytical Fourier cosine (COS) formula for the inverse Fourier transform of continuous probability distributions can be extended to discrete probability…
This paper presents a fast "two-dimensional Fourier Continuation" (2D-FC) method for the construction of biperiodic extensions of smooth, non-periodic functions defined over general two-dimensional (2D) domains, including domains with…
Despite the increasing prevalence of rotating-style capture (e.g., surveillance cameras), conventional stereo rectification techniques frequently fail due to the rotation-dominant motion and small baseline between views. In this paper, we…
In this paper, we introduce the fractional Fourier series on the fractional torus and study some basic facts of fractional Fourier series, such as fractional convolution and fractional approximation. Meanwhile, fractional Fourier inversion…
Data augmentation in feature space is effective to increase data diversity. Previous methods assume that different classes have the same covariance in their feature distributions. Thus, feature transform between different classes is…
A collection of algorithms is described for numerically computing with smooth functions defined on the unit disk. Low rank approximations to functions in polar geometries are formed by synthesizing the disk analogue of the double Fourier…
We consider finite approximations of a fractal generated by an iterated function system of affine transformations on $\mathbb{R}^d$ as a discrete set of data points. Considering a signal supported on this finite approximation, we propose a…
In some applications, one is interested in reconstructing a function $f$ from its Fourier series coefficients. The problem is that the Fourier series is slowly convergent if the function is non-periodic, or is non-smooth. In this paper, we…