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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 consider a method for fast numerical computation of the Fourier transform of a slowly decaying function with given accuracy in given ranges of the frequency. In these decades, some useful formulas for the Fourier transform…
The Fast Fourier Transform (FFT) is an algorithm of paramount importance in signal processing as it allows to apply the Fourier transform in O(n log n) instead of O(n 2) arithmetic operations. Graph Signal Processing (GSP) is a recent…
A new generalized function space in which all Gelfand-Shilov classes $S^{\prime 0}_\alpha$ ($\alpha>1$) of analytic functionals are embedded is introduced. This space of {\it ultrafunctionals} does not possess a natural nontrivial topology…
In this paper, we consider the problem of reconstructing piecewise smooth functions to high accuracy from nonuniform samples of their Fourier transform. We use the framework of nonuniform generalized sampling (NUGS) to do this, and to…
Continuous symmetries are fundamental to many scientific and learning problems, yet they are often unknown a priori. Existing symmetry discovery approaches typically search directly in the space of transformation generators or rely on…
This article introduces an effective generalization of the polar flavor of the Fourier Theorem based on a new method of analysis. Under the premises of the new theory an ample class of functions become viable as bases, with the further…
Fix $n \geq 2$ an integer, and $F$ be a totally real number field. We reduce the shifted convolution problem for $L$-function coefficients of $\operatorname{GL}_n({\bf{A}}_F)$-automorphic forms to the better-understood setting of…
On-going or soon to come cosmological large-scale structure surveys such as DESI, SPHEREx, Euclid, or the High-Latitude Spectroscopic Survey of the Nancy Grace Roman Space Telescope promise unprecedented measurement of the clustering of…
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…
We present a simple and fast algorithm for the computation of the coefficients of the expansion of a function f(cos u) in ultraspherical (Gegenbauer) polynomials. We prove that these coefficients coincide with the Fourier coefficients of an…
The $N$-point discrete Fourier transform (DFT) is a cornerstone for several signal processing applications. Many of these applications operate in real-time, making the computational complexity of the DFT a critical performance indicator to…
We formulate and derive a generalization of an orthogonal rational-function basis for spectral expansions over the infinite or semi-infinite interval. The original functions, first presented by Wiener are a mapping and weighting of the…
To a good approximation, on large cosmological scales the evolved two-point correlation function of biased tracers is related to the initial one by a convolution. For Gaussian initial conditions, the smearing kernel is Gaussian, so if the…
We construct operator analogues of Hermite functions which form an orthonormal basis for the Hilbert space $ \mathcal{S}_2$ of Hilbert-Schmidt operators on $ L^2(\R^n).$ We use this orthonormal basis to define Fourier transform on $…
We develop the uniform sparse Fast Fourier Transform (usFFT), an efficient, non-intrusive, adaptive algorithm for the solution of elliptic partial differential equations with random coefficients. The algorithm is an adaption of the sparse…
Object orientation provides a flexible framework for the implementation of the convolution of arbitrary distributions of real-valued random variables. We discuss an algorithm which is based on the discrete Fourier transformation (DFT) and…
This paper considers fast algorithms for operations on linearized polynomials. We propose a new multiplication algorithm for skew polynomials (a generalization of linearized polynomials) which has sub-quadratic complexity in the polynomial…
The fast computation of the Gauss hypergeometric function 2F1 with all its parameters complex is a difficult task. Although the 2F1 function verifies numerous analytical properties involving power series expansions whose implementation is…
The crossed channels of generalized reaction \gamma N - \gamma N have been considered. The transformation coefficients from the independent helicity amplitudes to the invariant functions are calculated. The explicit expressions for…