Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1412.1050

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

Emanuel Carneiro·Felipe Gonçalves
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

math.CA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Emanuel Carneiro, Felipe Gonçalves. 2014-12-02. Extremal problems in de Branges spaces: the case of truncated and odd functions. https://doi.org/10.1007/s00209-015-1411-1

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the log-concavity of the composite Bessel function $x^{\alpha}J_{\nu }\left( \beta x^{\gamma}\right) $

For a twice differentiable function $f:\left( a,b\right) \rightarrow \mathbb{R}$ define $v\left( f\right) =f^{\prime}f^{\prime}-f^{\prime\prime }f.$ It is well known that the positivity of $v\left( f\right) $ implies that the function $\left\vert f\right\vert $ is strictly log-concave on each subinterval which does not contain zeros of $f.$ In this paper we provide criteria for the positivity of $v\left( F\right) $ for the composite Bessel function $F\left( x\right) =J_{\alpha,\beta,\gamma,\nu}\left( x\right) :=x^{\alpha}J_{\nu}\left( \beta x^{\gamma}\right) $ for positive numbers $\beta$ and $\gamma$ and real numbers $\alpha$ and $\nu.$

math.CA↗
arXiv2026

Riesz capacity ratios with negative exponents

We investigate sharp inequalities for ratios of Riesz capacities with negative exponents by combining computational experiments with rigorous analysis. For finite subsets of the line, we prove positivity of equilibrium masses when $-1<p<0$, enabling numerical tests of conjectured extremal ratios. In the plane, comparisons of the disk with regular polygon vertex sets reveal a cascade of transitions among the tested competitors and suggest a precise conjecture for the equilibrium measure of odd polygons, for which we give a partial proof. Numerical intersections of equality curves show that the regions where these sets outperform the disk are not simply nested. Similar numerical intersections occur in three dimensions between the regular-simplex equality curve and those of explicit five-point and six-point configurations. Motivated by the dimensional dependence of these comparisons, we prove that for each fixed $p<-2<q<0$, the regular simplex has a larger capacity ratio than the ball in all sufficiently large dimensions. Accompanying Python and Mathematica code supports reproduction and further testing of the conjectures.

math.CA↗
arXiv2026

Shorter proof of dimension-free $L^p$ estimates for maximal Riesz transforms

We provide a shorter and more direct proof of $L^p$ estimates for maximal Riesz transforms (of an arbitrary order) in terms of the corresponding Riesz transforms, with a constant independent of the dimension of the Euclidean space $\mathbb R^d$. This result was originally proved by Mateu, Orobitg, P\'erez and Verdera with a constant depending on the dimension, and improved to a dimension-free inequality by Kucharski, Wr\'obel and Zienkiewicz.

math.CA↗