Related papers: Change of variable and the rapidity of decrease of…
We investigate the behaviour of the reduction type and related invariants of curves in families of curves over a discretely valued field. By a family, we will mean a set of curves obtained by perturbing the coefficients of the defining…
In the present article the author extends the Fourier transform to a more general class of functions; First to power-law functions with integer and half-integer exponents then to the widely used quantum statistics function (Fermi-Dirac and…
These are notes of a talk based on the work arXiv:1212.3630 joint with A. Aizenbud. Let V be a finite-dimensional vector space over a local field F of characteristic 0. Let f be a function on V of the form $f(x)= \psi (P(x))$, where P is a…
Ap\'ery's remarkable discovery of rapidly converging continued fractions with small coefficients for $\zeta(2)$ and $\zeta(3)$ has led to a flurry of important activity in an incredible variety of different directions. Our purpose is to…
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…
In this paper we consider Fourier transform techniques to efficiently compute the Value-at-Risk and the Conditional Value-at-Risk of an arbitrary loss random variable, characterized by having a computable generalized characteristic…
The Fourier Transform is one of the most important linear transformations used in science and engineering. Cooley and Tukey's Fast Fourier Transform (FFT) from 1964 is a method for computing this transformation in time $O(n\log n)$.…
We study the short-time Fourier transform on the space $\mathcal{K}_{1}'(\mathbb{R}^n)$ of distributions of exponential type. We give characterizations of $\mathcal{K}_{1}'(\mathbb{R}^n)$ and some of its subspaces in terms of modulation…
A rapid transformation is derived between spherical harmonic expansions and their analogues in a bivariate Fourier series. The change of basis is described in two steps: firstly, expansions in normalized associated Legendre functions of all…
We systematically find conditions which yield locally uniform convergence in the Fourier inversion formula in one and higher dimensions. We apply the gained knowledge to the complex inversion formula of the Laplace transform to extend known…
The well-known Bohr--P\'al theorem asserts that for every continuous real-valued function $f$ on the circle $\mathbb T$ there exists a change of variable, i.e., a homeomorphism $h$ of $\mathbb T$ onto itself, such that the Fourier series of…
This paper presents a generalized flux-corrected transport (FCT) algorithm, which is shown to be total variation diminishing under some conditions. The new algorithm has improved properties from the standpoint of use and analysis. Results…
The classification of natural exponential families started with the paper \cite {Morri} where Carl Morris unifies six very familiar families by the fact that their variance functions are polynomials of degree less or equal to two. Extension…
We consider the algebras $M_p$ of Fourier multipliers and show that every bounded continuous function $f$ on $\mathbb R^d$ can be transformed by an appropriate homeomorphic change of variable into a function that belongs to $M_p(\mathbb…
In this paper we study the variance of the Euler totient function (normalized to $\varphi(n)/n$) in the integers $\mathbb{Z}$ and in the polynomial ring $\mathbb{F}_q[T]$ over a finite field $\mathbb{F}_q$. It turns out that in…
We study the random sampling of the short-time Fourier transform of functions that are localized in a compact region in the time-frequency plane. We follow the approach introduced by Bass and Gr\"ochenig for band-limited functions, and show…
Several approaches to the formulation of a fractional theory of calculus of "variable order" have appeared in the literature over the years. Unfortunately, most of these proposals lack a rigorous mathematical framework. We consider an…
We consider non oscillatory functions and prove an everywhere Fourier Inversion Theorem for functions of very moderate decrease. The proofs rely on some ideas in nonstandard analysis.
Formul{\ae} are derived for expressing Cauchy and Hilbert transforms of a function $f$ in terms of Cauchy and Hilbert transforms of $f(x^r)$. When $r$ is an integer, this corresponds to evaluating the Cauchy transform of $f(x^r)$ at all…
The Fourier transform of a bounded measurable function, $f$, on the real line is shown to be the second distributional derivative of a H\"older continuous function. The Fourier transform is written as the difference of $\int_{-1}^1…