Related papers: Nonequispaced fast Fourier transforms for bandlimi…
In this paper, we introduce an algorithm that provides approximate solutions to semi-linear ordinary differential equations with highly oscillatory solutions, which, after an appropriate change of variables, can be rewritten as…
We derive and showcase a novel approach to approximating Fourier transforms in higher dimensions, focusing specifically on the case of 2D radially concentrated ('ring-like') functions. We first reduce the problem to that of evaluating the…
The problem of approximately computing the $k$ dominant Fourier coefficients of a vector $X$ quickly, and using few samples in time domain, is known as the Sparse Fourier Transform (sparse FFT) problem. A long line of work on the sparse FFT…
The convolution potential arises in a wide variety of application areas, and its efficient and accurate evaluation encounters three challenges: singularity, nonlocality and anisotropy. We introduce a fast algorithm based on a far-field…
An effective means to approximate an analytic, nonperiodic function on a bounded interval is by using a Fourier series on a larger domain. When constructed appropriately, this so-called Fourier extension is known to converge geometrically…
An integrated photonic circuit architecture to perform a modified-convolution operation based on the Discrete Fractional Fourier Transform (DFrFT) is introduced. This is accomplished by utilizing two nonuniformly-coupled waveguide lattices…
Many interesting and fundamentally practical optimization problems, ranging from optics, to signal processing, to radar and acoustics, involve constraints on the Fourier transform of a function. It is well-known that the {\em fast Fourier…
We introduce a fast algorithm for computing volume potentials - that is, the convolution of a translation invariant, free-space Green's function with a compactly supported source distribution defined on a uniform grid. The algorithm relies…
In this paper we explain how to use the Fast Fourier Transform (FFT) to solve partial differential equations (PDEs). We start by defining appropriate discrete domains in coordinate and frequency domains. Then describe the main limitation of…
This work extends Favard-type spectral representations for banded matrices $T$ beyond the bounded setting. It assumes that, for every $N\in\mathbb N_0$, there exists a shift $s_N\ge 0$ such that the shifted truncation $A_N:= T^{[N]}+s_N…
Quantum Fourier Transform (QFT) plays a principal role in the development of efficient quantum algorithms. Since the number of quantum bits that can currently built is limited, while many quantum technologies are inherently three- (or more)…
The Fractional Fourier Transform (FrFT) has widespread applications in areas like signal analysis, Fourier optics, diffraction theory, etc. The Holomorphic Fractional Fourier Transform (HFrFT) proposed in the present paper may be used in…
In this paper, we develop a continual analog of decomposition over orthogonal bases in spaces generated by equidistant shifts of a single function. By doing so, we obtain an explicit expression for best approximation by spaces of shifts in…
We propose the use of low bit-depth Sigma-Delta and distributed noise-shaping methods for quantizing the Random Fourier features (RFFs) associated with shift-invariant kernels. We prove that our quantized RFFs -- even in the case of $1$-bit…
We establish weighted $L^p$-Fourier-extension estimates for $O(N-k) \times O(k)$-invariant functions defined on the unit sphere $\mathbb{S}^{N-1}$, allowing for exponents $p$ below the Stein-Tomas critical exponent $\frac{2(N+1)}{N-1}$.…
Given a time series vector, how can we efficiently compute a specified part of Fourier coefficients? Fast Fourier transform (FFT) is a widely used algorithm that computes the discrete Fourier transform in many machine learning applications.…
In this paper, we provide a solution to the open problem of computing the Fourier transform of a binary function defined over $n$-bit vectors taking $m$-bit vector values. In particular, we introduce the two-modular Fourier transform (TMFT)…
In the presence of additive Gaussian noise, the statistics of the nonlinear Fourier transform (NFT) of a pulse are not yet completely known in closed form. In this paper, we propose a novel approach to study this problem. Our contributions…
An aperiodic (low frequency) spectrum may originate from the error term in the mean value of an arithmetical function such as M\"obius function or Mangoldt function, which are coding sequences for prime numbers. In the discrete Fourier…
The performance of multivariate kernel density estimation (KDE) depends strongly on the choice of bandwidth matrix. The high computational cost required for its estimation provides a big motivation to develop fast and accurate methods. One…