arXiv subjects
Publications and source records attributed to Felipe Gonçalves.
In this note we study the eigenvalue problem for a quadratic form associated with Strichartz estimates for the Schrödinger equation, proving in particular a sharp Strichartz inequality for the case of odd initial data. We also describe an alternative method that is applicable to a wider class of matrix problems.
Assuming the Riemann hypothesis we establish explicit bounds for the modulus of the log-derivative of Riemann's zeta-function in the critical strip.
We investigate the convergence of entire Lagrange interpolations and of Hermite interpolations of exponential type in weighted $L^p$-spaces on the real line. The weights are reciprocals of entire functions and depend on the type and may be viewed as smoothed versions of a target weight. The convergence statements are obtained from Marcinkiewicz inequalities with constants proportional to the type. For the special case of power weights we recover results of Rahman and Grozev and of Lubinsky.
We show that, possibly after a compactification of spacetime, constant functions are local maximizers of the Tomas-Stein adjoint Fourier restriction inequality for the cone and paraboloid in every dimension, and for the sphere in dimension up to 60. For the cone and paraboloid we work from the PDE framework, which enables the use of the Penrose and the Lens transformations, which map the conjectured optimal functions into constants.
In this note we develop a linear programming framework to produce upper and lower bounds for the lonely runner problem.
The sign uncertainty principle of Bourgain, Clozel & Kahane asserts that if a function $f:\mathbb{R}^d\to \mathbb{R}$ and its Fourier transform $\widehat{f}$ are nonpositive at the origin and not identically zero, then they cannot both be nonnegative outside an arbitrarily small neighborhood of the origin. In this article, we establish some equivalent formulations of the sign uncertainty principle, and in particular prove that minimizing sequences exist within the Schwartz class when $d=1$. We further address a complementary sign uncertainty principle, and show that corresponding near-minimizers concentrate a universal proportion of their positive mass near the origin in all dimensions.
In this paper we study the distribution of the non-trivial zeros of the Riemann zeta-function $ζ(s)$ (and other L-functions) using Montgomery's pair correlation approach. We use semidefinite programming to improve upon numerous asymptotic bounds in the theory of $ζ(s)$, including the proportion of distinct zeros, counts of small gaps between zeros, and sums involving multiplicities of zeros.
We prove an optimal bound in twelve dimensions for the uncertainty principle of Bourgain, Clozel, and Kahane. Suppose $f \colon \mathbb{R}^{12} \to \mathbb{R}$ is an integrable function that is not identically zero. Normalize its Fourier transform $\widehat{f}$ by $\widehat{f}(ξ) = \int_{\mathbb{R}^d} f(x)e^{-2πi \langle x, ξ\rangle}\, dx$, and suppose $\widehat{f}$ is real-valued and integrable. We show that if $f(0) \le 0$, $\widehat{f}(0) \le 0$, $f(x) \ge 0$ for $|x| \ge r_1$, and $\widehat{f}(ξ) \ge 0$ for $|ξ| \ge r_2$, then $r_1r_2 \ge 2$, and this bound is sharp. The construction of a function attaining the bound is based on Viazovska's modular form techniques, and its optimality follows from the existence of the Eisenstein series $E_6$. No sharp bound is known, or even conjectured, in any other dimension. We also develop a connection with the linear programming bound of Cohn and Elkies, which lets us generalize the sign pattern of $f$ and $\widehat{f}$ to develop a complementary uncertainty principle. This generalization unites the uncertainty principle with the linear programming bound as aspects of a broader theory.
We establish a universality law for sequences of functions $\{w_n\}_{n \in \mathbb{N}}$ satisfying a form of WKB approximation on compact intervals. This includes eigenfunctions of generic Schrödinger operators, as well as Laguerre and Chebyshev polynomials. Given two distinct points $x, y \in \mathbb{R}$, we ask how often do $w_n(x)$ and $w_n(y)$ have the same sign. Asymptotically, one would expect this to be true half the time, but this turns out to not always be the case. Under certain natural assumptions, we prove that, for all $x \neq y$, $$ \frac{1}{3} \leq \lim_{N \to \infty} \frac{1}{N} \# \left\{0 \leq n < N: \mathrm{sgn}(w_n(x)) = \mathrm{sgn}(w_n(y)) \right\} \leq \frac{2}{3}, $$ and that these bounds are optimal, and can be attained. Our methods extend to other questions of similar flavor and we also discuss a number of open problems.
We prove a sharpened version of the Strichartz inequality for radial solutions of the Schrödinger equation in $\mathbb{R}^2\times \mathbb{R}$. We establish an improved upper bound for functions that nearly extremize the inequality, with a negative second term that measures the distance from the initial data to Gaussians.
We study the Beurling-Selberg problem of finding band-limited $L^1$-functions that lie below the indicator function of an Euclidean ball. We compute the critical radius of the support of the Fourier transform for which such construction can have a positive integral.
In this paper we study sharp estimates for the Schrödinger operator via the framework of orthogonal polynomials. We use spherical harmonics and Gegenbauer polynomials to prove a new weighted inequality for the Schrödinger equation that is maximized by radial functions. We use Hermite and Laguerre polynomial expansions to produce sharp Strichartz estimates for even exponents. In particular, for radial initial data in dimension 2, we establish an interesting connection of the Strichartz norm with a combinatorial problem about words with four letters.
We investigate necessary and sufficient conditions under which entire functions in de Branges spaces can be recovered from function values and values of derivatives. Our main focus is on spaces with a structure function whose logarithmic derivative is bounded in the upper half-plane.
We study a recent result of Bourgain, Clozel and Kahane, a version of which states that a sufficiently nice function $f:\mathbb{R} \rightarrow \mathbb{R}$ that coincides with its Fourier transform and vanishes at the origin has a root in the interval $(c, \infty)$, where the optimal $c$ satisfies $0.41 \leq c \leq 0.64$. A similar result holds in higher dimensions. We improve the one-dimensional result to $0.45 \leq c \leq 0.594$, and the lower bound in higher dimensions. We also prove that extremizers exist, and have infinitely many double roots. With this purpose in mind, we establish a new structure statement about Hermite polynomials which relates their pointwise evaluation to linear flows on the torus, and applies to other families of orthogonal polynomials as well.
We prove an analogue of the Central Limit Theorem for operators. For every operator $K$ defined on $\mathbb{C}[x]$ we construct a sequence of operators $K_N$ defined on $\mathbb{C}[x_1,...,x_N]$ and demonstrate that, under certain orthogonality conditions, this sequence converges in a weak sense to an unique operator $\mathcal{C}$. We show that this operator $\mathcal{C}$ is a member of a family of operators $\mathfrak{C}$ that we call {\it Centered Gaussian Operators} and which coincides with the family of operators given by a centered Gaussian Kernel. Inspired in the approximation method used by Beckner in [W. Beckner, Inequalities in Fourier Analysis, Annals of Mathematics, 102 (1975), 159-182] to prove the sharp form of the Hausdorff-Young inequality, the present article shows that Beckner's method is a special case of a general approximation method for operators. In particular, we characterize the Hermite semi-group as the family of Centered Gaussian Operators associated with any semi-group of operators.
The purpose of this paper is to prove an interpolation formula involving derivatives for entire functions of exponential type. We extend the interpolation formula derived by J. Vaaler in [37, Theorem 9] to general $L^p$ de Branges spaces. We extensively use techniques from de Branges' theory of Hilbert spaces of entire functions as developed in [6], but a crucial passage involves the Hilbert-type inequalities as derived in [15]. We give applications to homogeneous spaces of entire functions that involve Bessel functions and we prove a uniqueness result for extremal one-sided band-limited approximations of radial functions in Euclidean spaces.
In this paper we find extremal one-sided approximations of exponential type for a class of truncated and odd functions with a certain exponential subordination. These approximations optimize the $L^1(\mathbb{R}, |E(x)|^{-2}dx)$-error, where $E$ is an arbitrary Hermite-Biehler entire function of bounded type in the upper half-plane. This extends the work of Holt and Vaaler [Duke Math. Journal 83 (1996), 203-247] for the signum function. We also provide periodic analogues of these results, finding optimal one-sided approximations by trigonometric polynomials of a given degree to a class of periodic functions with exponential subordination. These extremal trigonometric polynomials optimize the $L^1(\mathbb{R}/\mathbb{Z}, d\vartheta)$-error, where $\vartheta$ is an arbitrary nontrivial measure on $\mathbb{R}/\mathbb{Z}$. The periodic results extend the work of Li and Vaaler [Indiana Univ. Math. J. 48, No. 1 (1999), 183-236], who considered this problem for the sawtooth function with respect to Jacobi measures. Our techniques are based on the theory of reproducing kernel Hilbert spaces (of entire functions and of polynomials) and on the construction of suitable interpolations with nodes at the zeros of Laguerre-Pólya functions.
We construct majorants and minorants of a Gaussian function in Euclidean space that have Fourier transforms supported in a box. The majorants that we construct are shown to be extremal and our minorants are shown to be asymptotically extremal as the sides of the box become uniformly large. We then adapt the Distribution and Gaussian Subordination methods of Carneiro-Littmann-Vaaler to the multidimensional setting to obtain majorants and minorants for a class of radial functions. Periodic analogues of the main results are proven and applications to Hilbert-type inequalities are given.