arXiv subjects
Publications and source records attributed to Mateus Sousa.
In this paper we present a criteria to obtain interpolations formulas in terms of the sequence $\left(\{T_n(f)(Nm)\}\}_{m\in\mathbb{Z}}\right)_{n=1}^N$, where $f$ are functions whose Fourier transform is supported in $[-1/2,1/2]$, and $T_n$ are certain Fourier multiplier operators. We also discuss applications and also prove that our results recover several classical formulas.
We investigate the concept of Pauli pairs and a discrete counterpart to it. In particular, we make substantial progress on the question of when a discrete Pauli pair is automatically a classical Pauli pair. Effectively, if one of the functions has space and frequency Gaussian decay, and one has that $|f| = |g|$ and $|\widehat{f}| = |\widehat{g}|$ on two sets which accumulate like suitable small multiples of $\sqrt{n}$ at infinity, then $|f| \equiv |g|$ and $|\widehat{f}| = |\widehat{g}|.$ Furthermore, we show that if one drops either the assumption that one of the functions has space-frequency decay or that the discrete sets accumulate at a high rate, then the desired property no longer holds. Our techniques are inspired by and directly connected to several recent results in the realm of Fourier uniqueness problems, and our results may be seen as a nonlinear generalization of those. As a consequence of said techniques, we are able to prove a sharp discrete version of Hardy's uncertainty principle.
In this note, we study maximizers for Fourier extension inequalities on the sphere. We prove that constant functions are local maximizers for the $L^p(\mathbb{S}^{d-1})$ to $L^p(\mathbb{R}^d)$ Fourier extension estimates in the same range of exponents $p$ for which they are global maximizers for the $L^2(\mathbb{S}^{d-1})$ to $L^p_{rad}L^2_{ang}(\mathbb{R}^d)$ mixed-norm Fourier extension inequalities. Moreover, in the case of low dimensions, we improve the range of exponents for which constant functions are known to be the unique global maximizers for the $L^2(\mathbb{S}^{d-1})$ to $L^p_{rad}L^2_{ang}(\mathbb{R}^d)$ mixed-norm Fourier extension estimate on the sphere, covering, for the case of dimensions $d=2,3$, the entire Stein-Tomas range. This is achieved by establishing novel hierarchies between certain weighted norms of Bessel functions.
We study two variations of the classical one-delta problem for entire functions of exponential type, known also as the Carathéodory--Fejér--Turán problem. The first variation imposes the additional requirement that the function is radially decreasing while the second one is a generalization which involves derivatives of the entire function. Various interesting inequalities, inspired by results due to Duffin and Schaeffer, Landau, and Hardy and Littlewood, are also established.
In this paper we address the problem of estimating the operator norm of the embeddings between multidimensional weighted Paley-Wiener spaces. These can be equivalently thought as Fourier uncertainty principles for bandlimited functions. By means of radial symmetrization mechanisms, we show that such problems can all be shifted to dimension one. We provide precise asymptotics in the general case and, in some particular situations, we are able to identify the sharp constants and characterize the extremizers. The sharp constant study is actually a consequence of a more general result we prove in the setup of de Branges spaces of entire functions, addressing the operator given by multiplication by $z^k$, $k \in \mathbb{N}$. Applications to sharp higher order Poincar\'{e} inequalities and other related extremal problems are discussed.
We employ functional analysis techniques in order to deduce that some classical and recent interpolation results in Fourier analysis can be suitably perturbed. As an application of our techniques, we obtain generalizations of Kadec's 1/4-theorem for interpolation formulae in the Paley-Wiener space both in the real and complex case, as well as a perturbation result on the recent Radchenko-Viazovska interpolation result and the Cohn-Kumar-Miller-Radchenko-Viazovska result for Fourier interpolation with derivatives in dimensions 8 and 24. We also provide several applications of the main results and techniques, all relating to recent contributions in interpolation formulae and uniqueness sets for the Fourier transform.
For $ξ= (ξ_1, ξ_2, \ldots, ξ_d) \in \mathbb{R}^d$ let $Q(ξ) := \sum_{j=1}^d σ_j ξ_j^2$ be a quadratic form with signs $σ_j \in \{\pm1\}$ not all equal. Let $S \subset \mathbb{R}^{d+1}$ be the hyperbolic paraboloid given by $S = \big\{(ξ, τ) \in \mathbb{R}^{d}\times \mathbb{R} \ : \ τ= Q(ξ)\big\}$. In this note we prove that Gaussians never extremize an $L^p(\mathbb{R}^d) \to L^{q}(\mathbb{R}^{d+1})$ Fourier extension inequality associated to this surface.
We prove, under certain conditions on $(α,β)$, that each Schwartz function $f$ such that $f(\pm n^α) = \hat{f}(\pm n^β) = 0, \forall n \ge 0$ must vanish identically, complementing a series of recent results involving uncertainty principles, such as the pointwise interpolation formulas by Radchenko and Viazovska and the Meyer-Guinnand construction of self-dual crystaline measures.
We prove that in dimensions $d \geq 3$, the non-endpoint, Lorentz-invariant $L^2 \to L^p$ adjoint Fourier restriction inequality on the $d$-dimensional hyperboloid $\mathbb{H}^d \subseteq \mathbb{R}^{d+1}$ possesses maximizers. The analogous result had been previously established in dimensions $d=1,2$ using the convolution structure of the inequality at the lower endpoint (an even integer); we obtain the generalization by using tools from bilinear restriction theory.
We prove that maximal operators of convolution type associated to smooth kernels are bounded in the homogeneous Hardy-Sobolev spaces $\dot{H}^{1,p}(\mathbb{R}^d)$ when $1/p < 1+1/d$. This range of exponents is sharp. As a by-product of the proof, we obtain similar results for the local Hardy-Sobolev spaces $\dot{h}^{1,p}(\mathbb{R}^d)$ in the same range of exponents.
Let $d\geq 2$ be an integer and let $2d/(d-1) < q \leq \infty$. In this paper we investigate the sharp form of the mixed norm Fourier extension inequality \begin{equation*} \big\|\widehat{fσ}\big\|_{L^q_{\rm rad}L^2_{\rm ang}(\mathbb{R}^d)} \leq {\bf C}_{d,q}\, \|f\|_{L^2(\mathbb{S}^{d-1},{\rm d}σ)}, \end{equation*} established by L. Vega in 1988. Letting $\mathcal{A}_d \subset (2d/(d-1), \infty]$ be the set of exponents for which the constant functions on $\mathbb{S}^{d-1}$ are the unique extremizers of this inequality, we show that: (i) $\mathcal{A}_d$ contains the even integers and $\infty$; (ii) $\mathcal{A}_d$ is an open set in the extended topology; (iii) $\mathcal{A}_d$ contains a neighborhood of infinity $(q_0(d), \infty]$ with $q_0(d) \leq \left(\tfrac{1}{2} + o(1)\right) d\log d$. In low dimensions we show that $q_0(2) \leq 6.76\,;\,q_0(3) \leq 5.45 \,;\, q_0(4) \leq 5.53 \,;\, q_0(5) \leq 6.07$. In particular, this breaks for the first time the even exponent barrier in sharp Fourier restriction theory. The crux of the matter in our approach is to establish a hierarchy between certain weighted norms of Bessel functions, a nontrivial question of independent interest within the theory of special functions.
The $L^2 \to L^p$ adjoint Fourier restriction inequality on the $d$-dimensional hyperboloid $\mathbb{H}^d \subset \mathbb{R}^{d+1}$ holds provided $6 \leq p < \infty$, if $d=1$, and $2(d+2)/d \leq p\leq 2(d+1)/(d-1)$, if $d\geq2$. Quilodrán recently found the values of the optimal constants in the endpoint cases $(d,p)\in\{(2,4),(2,6),(3,4)\}$ and showed that the inequality does not have extremizers in these cases. In this paper we answer two questions posed by Quilodrán, namely: (i) we find the explicit value of the optimal constant in the endpoint case $(d,p) = (1,6)$ (the remaining endpoint for which $p$ is an even integer) and show that there are no extremizers in this case; and (ii) we establish the existence of extremizers in all non-endpoint cases in dimensions $d \in \{1,2\}$. This completes the qualitative description of this problem in low dimensions.
In this paper we establish that several maximal operators of convolution type, associated to elliptic and parabolic equations, are variation-diminishing. Our study considers maximal operators on the Euclidean space $\mathbb{R}^d$, on the torus $\mathbb{T}^d$ and on the sphere $\mathbb{S}^d$. The crucial regularity property that these maximal functions share is that they are subharmonic in the corresponding detachment sets.