arXiv subjects

Emanuel Carneiro

Publications and source records attributed to Emanuel Carneiro.

54 records · Page 3Linked to original sources

arXiv2014

Extremal problems in de Branges spaces: the case of truncated and odd functions

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.

math.CA↗

arXiv2014

Extremal functions in de Branges and Euclidean spaces

In this work we obtain optimal majorants and minorants of exponential type for a wide class of radial functions on $\mathbb{R}^N$. These extremal functions minimize the $L^1(\mathbb{R}^N, |x|^{2ν+ 2 - N}dx)$-distance to the original function, where $ν>-1$ is a free parameter. To achieve this result we develop new interpolation tools to solve an associated extremal problem for the exponential function $\mathcal{F}_λ(x) = e^{-λ|x|}$, where $λ>0$, in the general framework of de Branges spaces of entire functions. We then specialize the construction to a particular family of homogeneous de Branges spaces to approach the multidimensional Euclidean case. Finally, we extend the result from the exponential function to a class of subordinated radial functions via integration on the parameter $λ>0$ against suitable measures. Applications of the results presented here include multidimensional versions of Hilbert-type inequalities, extremal one-sided approximations by trigonometric polynomials for a class of even periodic functions and extremal one-sided approximations by polynomials for a class of functions on the sphere $\mathbb{S}^{N-1}$ with an axis of symmetry.

math.CA↗

arXiv2014

Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function

Montgomery's pair correlation conjecture predicts the asymptotic behavior of the function $N(T,β)$ defined to be the number of pairs $γ$ and $γ'$ of ordinates of nontrivial zeros of the Riemann zeta-function satisfying $0<γ,γ'\leq T$ and $0 < γ'-γ\leq 2πβ/\log T$ as $T\to \infty$. In this paper, assuming the Riemann hypothesis, we prove upper and lower bounds for $N(T,β)$, for all $β>0$, using Montgomery's formula and some extremal functions of exponential type. These functions are optimal in the sense that they majorize and minorize the characteristic function of the interval $[-β, β]$ in a way to minimize the $L^1\big(\mathbb{R}, \big\{1 - \big(\frac{\sin πx}{πx}\big)^2 \big\}\,dx\big)$-error. We give a complete solution for this extremal problem using the framework of reproducing kernel Hilbert spaces of entire functions. This extends previous work by P. X. Gallagher in 1985, where the case $β\in \frac12 \mathbb{N}$ was considered using non-extremal majorants and minorants.

math.NT↗

arXiv2014

Some sharp restriction inequalities on the sphere

In this paper we find the sharp forms and characterize the complex-valued extremizers of the adjoint Fourier restriction inequalities on the sphere $$\big\|\widehat{f σ}\big\|_{L^{p}(\mathbb{R}^{d})} \lesssim \|f\|_{L^{q}(\mathbb{S}^{d-1},σ)}$$ in the cases $(d,p,q) = (d,2k, q)$ with $d,k \in \mathbb{N}$ and $q\in \mathbb{R}^+ \cup \{\infty\}$ satisfying: (a) $k = 2$, $q \geq 2$ and $3 \leq d\leq 7$; (b) $k = 2$, $q \geq 4$ and $d \geq 8$; (c) $k \geq 3$, $q \geq 2k$ and $d \geq 2$. We also prove a sharp multilinear weighted restriction inequality, with weight related to the $k$-fold convolution of the surface measure.

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arXiv2013

Entire approximations for a class of truncated and odd functions

We solve the problem of finding optimal entire approximations of prescribed exponential type (unrestricted, majorant and minorant) for a class of truncated and odd functions with a shifted exponential subordination, minimizing the $L^1(\R)$-error. The class considered here includes new examples such as the truncated logarithm and truncated shifted power functions. This paper is the counterpart of the works of Carneiro and Vaaler (Some extremal functions in Fourier analysis, Part II in Trans. Amer. Math. Soc. 362 (2010), 5803-5843; Part III in Constr. Approx. 31, No. 2 (2010), 259--288), where the analogous problem for even functions was treated.

math.CA↗

arXiv2013

Bounding $S(t)$ and $S_1(t)$ on the Riemann hypothesis

Let $πS(t)$ denote the argument of the Riemann zeta-function, $ζ(s)$, at the point $s=\frac{1}{2}+it$. Assuming the Riemann hypothesis, we present two proofs of the bound $$ |S(t)| \leq \left(\tfrac{1}{4} + o(1) \right)\tfrac{\log t}{\log \log t} $$ for large $t$. This improves a result of Goldston and Gonek by a factor of 2. The first method consists in bounding the auxiliary function $S_1(t) = \int_0^{t} S(u) du$ using extremal functions constructed by Carneiro, Littmann and Vaaler. We then relate the size of $S(t)$ to the size of the functions $S_1(t\pm h)-S_1(t)$ when $h\asymp 1/\log\log t$. The alternative approach bounds $S(t)$ directly, relying on the solution of the Beurling-Selberg extremal problem for the odd function $f(x) = \arctan\left(\tfrac{1}{x}\right) - \tfrac{x}{1 + x^2}$. This draws upon recent work by Carneiro and Littmann.

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arXiv2013

On the variation of maximal operators of convolution type

In this paper we study the regularity properties of two maximal operators of convolution type: the heat flow maximal operator (associated to the Gauss kernel) and the Poisson maximal operator (associated to the Poisson kernel). In dimension $d=1$ we prove that these maximal operators do not increase the $L^p$-variation of a function for any $p \geq 1$, while in dimensions $d>1$ we obtain the corresponding results for the $L^2$-variation. Similar results are proved for the discrete versions of these operators.

math.AP↗

arXiv2013

On the endpoint regularity of discrete maximal operators

Given a discrete function $f:\Z^d \to \R$ we consider the maximal operator $$Mf(\vec{n}) = \sup_{r\geq0} \frac{1}{N(r)} \sum_{\vec{m} \in \barΩ_r} \big|f(\vec{n} + \vec{m})\big|,$$ where $\big\{\barΩ_r\big\}_{r \geq 0}$ are dilations of a convex set $Ω$ (open, bounded and with Lipschitz boudary) containing the origin and $N(r)$ is the number of lattice points inside $\barΩ_r$. We prove here that the operator $f \mapsto \nabla M f$ is bounded and continuous from $l^1(\Z^d)$ to $l^1(\Z^d)$. We also prove the same result for the non-centered version of this discrete maximal operator.

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arXiv2012

Bandlimited approximations to the truncated Gaussian and applications

In this paper we extend the theory of optimal approximations of functions $f: \R \to \R$ in the $L^1(\R)$-metric by entire functions of prescribed exponential type (bandlimited functions). We solve this problem for the truncated and the odd Gaussians using explicit integral representations and fine properties of truncated theta functions obtained via the maximum principle for the heat operator. As applications, we recover most of the previously known examples in the literature and further extend the class of truncated and odd functions for which this extremal problem can be solved, by integration on the free parameter and the use of tempered distribution arguments. This is the counterpart of the work \cite{CLV}, where the case of even functions is treated.

math.CA↗

arXiv2011

Gaussian Subordination for the Beurling-Selberg Extremal Problem

We determine extremal entire functions for the problem of majorizing, minorizing, and approximating the Gaussian function $e^{-πλx^2}$ by entire functions of exponential type. This leads to the solution of analogous extremal problems for a wide class of even functions that includes most of the previously known examples (for instance \cite{CV2}, \cite{CV3}, \cite{GV} and \cite{Lit}), plus a variety of new interesting functions such as $|x|^α$ for $-1 < α$; \,$\log \,\bigl((x^2 + α^2)/(x^2 + β^2)\bigr)$, for $0 \leq α< β$;\, $\log\bigl(x^2 + α^2\bigr)$; and $x^{2n} \log x^2$\,, for $n \in \N$. Further applications to number theory include optimal approximations of theta functions by trigonometric polynomials and optimal bounds for certain Hilbert-type inequalities related to the discrete Hardy-Littlewood-Sobolev inequality in dimension one.

math.CA↗

arXiv2010

Bounding $ζ(s)$ in the critical strip

Assuming the Riemann Hypothesis, we make use of the recently discovered \cite{CLV} extremal majorants and minorants of prescribed exponential type for the function $\log\left(\tfrac{4 + x^2}{(α-1/2)^2 + x^2}\right)$ to find upper and lower bounds with explicit constants for $\log|ζ(α+ it)|$ in the critical strip, extending the work of Chandee and Soundararajan \cite{CS}.

math.NT↗

arXiv2010

Convolution inequalities for the Boltzmann collision operator

We study integrability properties of a general version of the Boltzmann collision operator for hard and soft potentials in $n$-dimensions. A reformulation of the collisional integrals allows us to write the weak form of the collision operator as a weighted convolution, where the weight is given by an operator invariant under rotations. Using a symmetrization technique in $L^p$ we prove a Young's inequality for hard potentials, which is sharp for Maxwell molecules in the $L^2$ case. Further, we find a new Hardy-Littlewood-Sobolev type of inequality for Boltzmann collision integrals with soft potentials. The same method extends to radially symmetric, non-increasing potentials that lie in some $L^{s}_{weak}$ or $L^{s}$. The method we use resembles a Brascamp, Lieb and Luttinger approach for multilinear weighted convolution inequalities and follows a weak formulation setting. Consequently, it is closely connected to the classical analysis of Young and Hardy-Littlewood-Sobolev inequalities. In all cases, the inequality constants are explicitly given by formulas depending on integrability conditions of the angular cross section (in the spirit of Grad cut-off). As an additional application of the technique we also obtain estimates with exponential weights for hard potentials in both conservative and dissipative interactions.

math.AP↗

arXiv2009

Estimates for the Boltzmann collision operator via radial symmetry and Fourier transform

We extend the $L^p$-theory of the Boltzmann collision operator by using classical techniques based in the Carleman representation and Fourier analysis, allied to new ideas that exploit the radial symmetry of this operator. We are then able to greatly simplify existent technical proofs in this theory, extend the range, and obtain explicit sharp constants in some convolution-like inequalities for the gain part of the Boltzmann collision operator.

math.AP↗

arXiv2008

Sharp approximations to the Bernoulli periodic functions by trigonometric polynomials

We obtain optimal trigonometric polynomials of a given degree $N$ that majorize, minorize and approximate in $L^1(\mathbb{R}/\mathbb{Z})$ the Bernoulli periodic functions. These are the periodic analogues of two works of F. Littmann that generalize a paper of J. Vaaler. As applications we provide the corresponding Erdös-Turán-type inequalities, approximations to other periodic functions and bounds for certain Hermitian forms.

math.CA↗

arXiv2008

Some extremal functions in Fourier analysis, III

We obtain the best approximation in $L^1(\R)$, by entire functions of exponential type, for a class of even functions that includes $e^{-λ|x|}$, where $λ>0$, $\log |x|$ and $|x|^α$, where $-1 < α< 1$. We also give periodic versions of these results where the approximating functions are trigonometric polynomials of bounded degree.

math.CA↗

arXiv2008

On the regularity of maximal operators

We study the regularity of the bilinear maximal operator when applied to Sobolev functions, proving that it maps $W^{1,p}(\mathbb{R}) \times W^{1,q}(\mathbb{R}) \to W^{1,r}(\mathbb{R})$ with $1 1$. We also investigate the almost everywhere and weak convergence under the action of the classical Hardy-Littlewood maximal operator, both in its global and local versions.

math.CA↗

arXiv2008

Some extremal functions in Fourier analysis, II

We obtain extremal majorants and minorants of exponential type for a class of even functions on $\R$ which includes $\log |x|$ and $|x|^α$, where $-1 < α< 1$. We also give periodic versions of these results in which the majorants and minorants are trigonometric polynomials of bounded degree. As applications we obtain optimal estimates for certain Hermitian forms, which include discrete analogues of the one dimensional Hardy-Littlewood-Sobolev inequalities. A further application provides an Erdös-Turán-type inequality that estimates the sup norm of algebraic polynomials on the unit disc in terms of power sums in the roots of the polynomials.

math.CA↗

arXiv2008

A sharp inequality for the Strichartz norm

Let $u:\R \times \R^n \to \C$ be the solution of the linear Schrödinger equation $iu_t + Δu =0$ with initial data $u(0,x) = f(x)$. In the first part of this paper we obtain a sharp inequality for the Strichartz norm $\|u(t,x)\|_{L^{2k}_tL^{2k}_x(\R \times\R^n)}$, where $k\in \Z$, $k \geq 2$ and $(n,k) \neq (1,2)$, that admits only Gaussian maximizers. As corollaries we obtain sharp forms of the classical Strichartz inequalities in low dimensions (works of Foschi and Hundertmark - Zharnitsky) and also sharp forms of some Sobolev-Strichartz inequalities. In the second part of the paper we express Foschi's sharp inequalities for the Schrödinger and wave equations in the broader setting of sharp restriction/extension estimates for the paraboloid and the cone.

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