Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDiscrete Uniqueness Sets for Functions with Spectral Gaps
60
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
It is well-known that entire functions whose spectrum belongs to a fixed bounded set $S$ admit real uniformly discrete uniqueness sets $Lambda$. We show that the same is true for much wider spaces of continuous functions. In particular, Sobolev spaces have this property whenever $S$ is a set of infinite measure having periodic gaps. The periodicity condition is crucial. For sets $S$ with randomly distributed gaps, we show that the uniformly discrete sets $Lambda$ satisfy a strong non-uniqueness property: Every discrete function $c(lambda)in l^2(Lambda)$ can be interpolated by an analytic $L^2$-function with spectrum in $S$.
rate research
Read More
This paper builds upon two key principles behind the Bourgain-Dyatlov quantitative uniqueness theorem for functions with Fourier transform supported in an Ahlfors regular set. We first provide a characterization of when a quantitative uniqueness theorem holds for functions with very quickly decaying Fourier transform, thereby providing an extension of the classical Paneah-Logvinenko-Sereda theorem. Secondly, we derive a transference result which converts a quantitative uniqueness theorem for functions with fast decaying Fourier transform to one for functions with Fourier transform supported on a fractal set. As well as recovering the result of Bourgain-Dyatlov, we obtain analogous uniqueness results for denser fractals.
We prove new $ell ^{p} (mathbb Z ^{d})$ bounds for discrete spherical averages in dimensions $ d geq 5$. We focus on the case of lacunary radii, first for general lacunary radii, and then for certain kinds of highly composite choices of radii. In particular, if $ A _{lambda } f $ is the spherical average of $ f$ over the discrete sphere of radius $ lambda $, we have begin{equation*} bigllVert sup _{k} lvert A _{lambda _k} f rvert bigrrVert _{ell ^{p} (mathbb Z ^{d})} lesssim lVert frVert _{ell ^{p} (mathbb Z ^{d})}, qquad tfrac{d-2} {d-3} < p leq tfrac{d} {d-2}, dgeq 5, end{equation*} for any lacunary sets of integers $ {lambda _k ^2 }$. We follow a style of argument from our prior paper, addressing the full supremum. The relevant maximal operator is decomposed into several parts; each part requires only one endpoint estimate.
For a function $fcolon [0,1]tomathbb R$, we consider the set $E(f)$ of points at which $f$ cuts the real axis. Given $fcolon [0,1]tomathbb R$ and a Cantor set $Dsubset [0,1]$ with ${0,1}subset D$, we obtain conditions equivalent to the conjunction $fin C[0,1]$ (or $fin C^infty [0,1]$) and $Dsubset E(f)$. This generalizes some ideas of Zabeti. We observe that, if $f$ is continuous, then $E(f)$ is a closed nowhere dense subset of $f^{-1}[{ 0}]$ where each $xin {0,1}cap E(f)$ is an accumulation point of $E(f)$. Our main result states that, for a closed nowhere dense set $Fsubset [0,1]$ with each $xin {0,1}cap E(f)$ being an accumulation point of $F$, there exists $fin C^infty [0,1]$ such that $F=E(f)$.
We consider inductive limits of weighted spaces of holomorphic functions in the unit ball of $mathbb C^n$. The relationship between sets of uniqueness, weakly sufficient sets and sampling sets in these spaces is studied. In particular, the equivalence of these sets, under general conditions of the weights, is obtained.
The Cholesky factorization of the moment matrix is applied to discrete orthogonal polynomials on the homogeneous lattice. In particular, semiclassical discrete orthogonal polynomials, which are built in terms of a discrete Pearson equation, are studied. The Laguerre-Freud structure semi-infinite matrix that models the shifts by $pm 1$ in the independent variable of the set of orthogonal polynomials is introduced. In the semiclassical case it is proven that this Laguerre-Freud matrix is banded. From the well known fact that moments of the semiclassical weights are logarithmic derivatives of generalized hypergeometric functions, it is shown how the contiguous relations for these hypergeometric functions translate as symmetries for the corresponding moment matrix. It is found that the 3D Nijhoff-Capel discrete Toda lattice describes the corresponding contiguous shifts for the squared norms of the orthogonal polynomials. The continuous Toda for these semiclassical discrete orthogonal polynomials is discussed and the compatibility equations are derived. It also shown that the Kadomtesev-Petvishvilii equation is connected to an adequate deformed semiclassical discrete weight, but in this case the deformation do not satisfy a Pearson equation.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
