arXiv subjects

Friedrich Littmann

Publications and source records attributed to Friedrich Littmann.

15 recordsLinked to original sources

arXiv2026

A Painlev\'e equation for the H\"ormander-Bernhardsson constant

The H\"ormander-Bernhardsson constant $\mathscr{C}$ is the sharp constant in $|f(0)|\le \mathscr{C} \|f\|_1$ for entire functions of exponential type $\le \pi$. We prove that $\mathscr{C} = 2\pi \theta_*^{-2}$ where $\theta_*$ is the least positive singularity of the regular solution $v$ with $v(0)=0$ of the cosh-Gordon equation $v_{\theta\theta} + v_\theta/\theta = \cosh(v)$. It is known that $\mathscr{C}$ is a scaling limit in $n$ from the analogous problem for polynomials of degree $\le n$. We reformulate the polynomial problem as a Pad\'e approximation problem at infinity. The associated matrix Riemann-Hilbert problem is analyzed by a Deift-Zhou steepest descent whose local parametrix is built from a Painlev\'e transcendent.

math.CA↗

arXiv2023

Monotone extremal functions and the weighted Hilbert's inequality

In this note we find optimal one-sided majorants of exponential type for the signum function subject to certain monotonicity conditions. As an application, we use these special functions to obtain a simple Fourier analysis proof of the (non-sharp) weighted Hilbert-Montgomery-Vaughan inequality.

math.CA↗

arXiv2022

Concentration inequalities for Paley-Wiener spaces

This article considers the question of how much of the mass of an element in a Paley-Wiener space can be concentracted on a given set. We seek bounds in terms of relative densities of the given set. We extend a result of Donoho and Logan from 1992 in one dimension and consider similar results in higher dimensions.

math.CA↗

arXiv2021

Mean convergence of entire interpolations in weighted space

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.

math.CA↗

arXiv2020

Weighted uniform convergence of entire Grünwald operators on the real line

We consider weighted uniform convergence of entire analogues of the Grünwald operator on the real line. The main result deals with convergence of entire interpolations of exponential type $τ>0$ at zeros of Bessel functions in spaces with homogeneous weights. We discuss extensions to Grünwald operators from de Branges spaces.

math.CA↗

arXiv2019

Conjugate Phase Retrieval in Paley-Wiener Space

We consider the problem of conjugate phase retrieval in Paley-Wiener space $PW_π$. The goal of conjugate phase retrieval is to recover a signal $f$ from the magnitudes of linear measurements up to unknown phase factor and unknown conjugate, meaning $f(t)$ and $\overline{f(t)}$ are not necessarily distinguishable from the available data. We show that conjugate phase retrieval can be accomplished in $PW_π$ by sampling only on the real line by using structured convolutions. We also show that conjugate phase retrieval can be accomplished in $PW_π$ by sampling both $f$ and $f^{\prime}$ only on the real line. Moreover, we demonstrate experimentally that the Gerchberg-Saxton method of alternating projections can accomplish the reconstruction from vectors that do conjugate phase retrieval in finite dimensional spaces. Finally, we show that generically, conjugate phase retrieval can be accomplished by sampling at three times the Nyquist rate, whereas phase retrieval requires sampling at four times the Nyquist rate.

cs.IT↗

arXiv2017

Interpolation Formulas With Derivatives in De Branges Spaces II

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.

math.CV↗

arXiv2016

Extremal Signatures

Let $E= A - iB$ be a Hermite-Biehler entire function of exponential type $τ/2$ where $A$ and $B$ are real entire, and consider $dμ(x) = dx/|E(x)|^2$. We show that the sign of the product $A B$ is an extremal signature for the space of functions of exponential type $τ$ with respect to the norm of $L^1(μ)$. This allows us to find best approximations by entire functions of exponential type $τ$ in $L^1(μ)$-norm to certain special functions (e.g., the Gaussian and the Poisson kernel).

math.CA↗

arXiv2016

Extremal functions with vanishing condition

We determine the optimal majorant $M^+$ and minorant $M^-$ of exponential type for the truncation of $x\mapsto (x^2+a^2)^{-1}$ with respect to general de Branges measures. We prove that \[ \int_\mathbb{R} (M^+ - M^-) |E(x)|^{-2}dx = \frac{1}{a^2 K(0,0)} \] where $K$ is the reproducing kernel for $\mathcal{H}(E)$. As an application we determine the optimal majorant and minorant for the Heaviside function that vanish at a fixed point $α= ia$ on the imaginary axis. We show that the difference of majorant and minorant has integral value $(πa - \tanh(πa))^{-1} πa$.

math.CA↗

arXiv2015

Extremal functions in de Branges and Euclidean spaces II

This paper presents the Gaussian subordination framework to generate optimal one-sided approximations to multidimensional real-valued functions by functions of prescribed exponential type. Such extremal problems date back to the works of Beurling and Selberg and provide a variety of applications in analysis and analytic number theory. Here we majorize and minorize (on $\mathbb{R}^N$) the Gaussian ${\bf x} \mapsto e^{-πλ|{\bf x}|^2}$, where $λ>0$ is a free parameter, by functions with distributional Fourier transforms supported on Euclidean balls, optimizing weighted $L^1$-errors. By integrating the parameter $λ$ against suitable measures, we solve the analogous problem for a wide class of radial functions. Applications to inequalities and periodic analogues are discussed. The constructions presented here rely on the theory of de Branges spaces of entire functions and on new interpolations tools derived from the theory of Laplace transforms 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↗

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↗

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↗