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Every set $Lambdasubset R$ such that the sum of $delta$-measures sitting at the points of $Lambda$ is a Fourier quasicrystal, is the zero set of an exponential polynomial with imaginary frequencies.
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By using computers to do experimental manipulations on Fourier series, we construct additional series with interesting properties. For example, we construct several series whose sums remain unchanged when the nth term is multiplied by sin(n)/n. One series with this property is this classic series for pi/4: pi/4 = 1 - 1/3 + 1/5 ... = 1*(sin(1)/1) - (1/3)*(sin(3)/3) + (1/5)*(sin(5)/5).... Another example is sum (n = 1 to infinity) of (sin(n)/n) = sum (n = 1 to infinity) of (sin(n)/n)^2 = (pi - 1)/2. This material should be accessible to undergraduates. This paper also includes a Mathematica package that makes it easy to calculate and graph the Fourier series of many types of functions.
In this paper, we present a different proof on the discrete Fourier restriction. The proof recovers Bourgains level set result on Strichartz estimates associated with Schrodinger equations on torus. Some sharp estimates on $L^{frac{2(d+2)}{d}}$ norm of certain exponential sums in higher dimensional cases are established. As an application, we show that some discrete multilinear maximal functions are bounded on $L^2(mathbb Z)$.
In this paper, we consider a discrete restriction associated with KdV equations. Some new Strichartz estimates are obtained. We also establish the local well-posedness for the periodic generalized Korteweg-de Vries equation with nonlinear term $ F(u)p_x u$ provided $Fin C^5$ and the initial data $phiin H^s$ with $s>1/2$.
In the present paper we develop the theory of minimization for energies with multivariate kernels, i.e. energies, in which pairwise interactions are replaced by interactions between triples or, more generally, $n$-tuples of particles. Such objects, which arise naturally in various fields, present subtle differences and complications when compared to the classical two-input case. We introduce appropriate analogues of conditionally positive definite kernels, establish a series of relevant results in potential theory, explore rotationally invariant energies on the sphere, and present a variety of interesting examples, in particular, some optimization problems in probabilistic geometry which are related to multivaria
We study Fourier multipliers on free group $mathbb{F}_infty$ associated with the first segment of the reduced words, and prove that they are completely bounded on the noncommutative $L^p$ spaces $L^p(hat{mathbb{F}}_infty)$ iff their restriction on $L^p(hat{mathbb{F}}_1)=L^p(mathbb{T})$ are completely bounded. As a consequence, every classical Mikhlin multiplier extends to a $L^p$ Fourier multiplier on free groups for all $1<p<infty$.
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