Related papers: Fourier transforms of bounded functions

In recent years there has been a growing interest in the fractional Fourier transform driven by its large number of applications. The literature in this field follows two main routes. On the one hand, the areas where the ordinary Fourier…

This is the direct continuation of the paper "Mapping properties of Fourier transforms" (arXiv:2112.04896) using the same notation as there without further explanations. It deals with continuous and compact mappings of the Fourier transform…

We first construct a space $\mathcal{W}\left( \mathbb{R}_{\text{c}} ^{n}\right) $ whose elements are test functions defined in $\mathbb{R} _{\text{c}}^{n}=\mathbb{R}^{n}\cup\left\{ \mathbf{\infty}\right\} ,$ the one point compactification…

If $f\in L^1({\mathbb R})$ it is proved that $\lim_{S\to\infty}\lVert f-f\ast D_S\rVert=0$, where $D_S(x)=\sin(Sx)/(\pi x)$ is the Dirichlet kernel and $\lVert f\rVert = \sup_{\alpha<\beta}|\int_{\alpha}^{\beta}f(x)\,dx|$ is the Alexiewicz…

The Fourier coefficient of a second order Eisenstein series is described as a shifted convolution sum. This description is used to obtain the spectral decomposition of and estimates for the shifted convolution sum.

We show a pointwise estimate for the Fourier transform on the line involving the number of times the function changes monotonicity. The contrapositive of the theorem may be used to find a lower bound to the number of local maxima of a…

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 order to compute the Fourier transform of a function $f$ on the real line numerically, one samples $f$ on a grid and then takes the discrete Fourier transform. We derive exact error estimates for this procedure in terms of the decay and…

In this note, we prove that the Fourier-Laplace transform of the typical function (i.e., generic in the sense of Baire category theorem) in the Schwartz class of the half-line, being analytic in the lower half of the complex plane, has…

For the Fourier transform $\mathcal{F}\mu$ of a general (non-trivial) self-similar measure $\mu$ on the real line $\mathbb{R}$, we prove a large deviation estimate \[ \lim_{c\to +0} \varlimsup_{t\to \infty}\frac{1}{t}\log…

A new integral identity for functions with continuous second partial derivatives is derived. It is shown that the value of any function f(r,t) at position r and time t is completely determined by its previous values at all other locations…

The classical Fourier transform is, in essence, a way to take data and extract components (in the form of complex exponentials) which are invariant under cyclic shifts. We consider a case in which the components must instead be invariant…

In this paper, we propose a numerical method of Fourier transform based on hyperfunction theory. In the proposed method, we compute analytic functions called the defining functions, which give the desired Fourier transform as a…

The discrete Fourier transform of the greatest common divisor is a multiplicative function that generalises both the gcd-sum function and Euler's totient function. On the one hand it is the Dirichlet convolution of the identity with…

We appeal to a complex q-Fourier transform as a generalization of the (real) one analyzed in [Milan J. Math. {\bf 76} (2008) 307]. By recourse to tempered ultra-distributions we are able to show that the q-Gaussian distribution can be…

This paper determines how to define a discretely implemented Fourier transform when analysing an observed spatial point process. To develop this transform we answer four questions; first what is the natural definition of a Fourier…

We introduce the directional short-time fractional Fourier transform (DSTFRFT) and prove an extended Parseval's identity and a reconstruction formula for it. We also investigate the continuity of both the directional short-time fractional…

A recent generalization of the Central Limit Theorem consistent with nonextensive statistical mechanics has been recently achieved through a generalized Fourier transform, noted $q$-Fourier transform. A representation formula for the…

This is a sequel to math.AG/0003009. Here we study identities for the Fourier transform of "elementary functions" over finite field containing "exponents" of monomial rational functions. It turns out that these identities are governed by…

We propose an alternative definition for a Tsallis entropy composition-inspired Fourier transform, which we call "$\tau_q$-Fourier transform". We comment about the underlying "covariance" on the set of algebraic fields that motivates its…