Gaussian Subordination for the Beurling-Selberg Extremal Problem

We determine extremal entire functions for the problem of majorizing, minorizing, and approximating the Gaussian function $e^{-\pi\lambda x^2}$ 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 $|x|^{\alpha}$ for $-1 < \alpha$; \,$\log \,\bigl((x^2 + \alpha^2)/(x^2 + \beta^2)\bigr)$, for $0 \leq \alpha < \beta$;\, $\log\bigl(x^2 + \alpha^2\bigr)$; 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

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 $N(T,\beta)$ defined to be the number of pairs $\gamma$ and $\gamma'$ of ordinates of nontrivial zeros of the Riemann zeta-function satisfying $0<\gamma,\gamma'\leq T$ and $0 < \gamma'-\gamma \leq 2\pi \beta/\log T$ as $T\to \infty$. In this paper, assuming the Riemann hypothesis, we prove upper and lower bounds for $N(T,\beta)$, for all $\beta >0$, 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 $[-\beta, \beta]$ in a way to minimize the $L^1\big(\mathbb{R}, \big\{1 - \big(\frac{\sin \pi x}{\pi x}\big)^2 \big\}\,dx\big)$-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 $\beta \in \frac12 \mathbb{N}$ was considered using non-extremal majorants and minorants.Comment: to appear in J. Reine Angew. Mat

Crystalline iron oxides stimulate methanogenic benzoate degradation in marine sediment- derived enrichment cultures

Elevated dissolved iron concentrations in the methanic zone are typical geochemical signatures of rapidly accumulating marine sediments. These sediments are often characterized by co-burial of iron oxides with recalcitrant aromatic organic matter of terrigenous origin. Thus far, iron oxides are predicted to either impede organic matter degradation, aiding its preservation, or identified to enhance organic carbon oxidation via direct electron transfer. Here, we investigated the effect of various iron oxide phases with differing crystallinity (magnetite, hematite, and lepidocrocite) during microbial degradation of the aromatic model compound benzoate in methanic sediments. In slurry incubations with magnetite or hematite, concurrent iron reduction, and methanogenesis were stimulated during accelerated benzoate degradation with methanogenesis as the dominant electron sink. In contrast, with lepidocrocite, benzoate degradation, and methanogenesis were inhibited. These observations were reproducible in sediment-free enrichments, even after five successive transfers. Genes involved in the complete degradation of benzoate were identified in multiple metagenome assembled genomes. Four previously unknown benzoate degraders of the genera Thermincola (Peptococcaceae, Firmicutes), Dethiobacter (Syntrophomonadaceae, Firmicutes), Deltaproteobacteria bacteria SG8_13 (Desulfosarcinaceae, Deltaproteobacteria), and Melioribacter (Melioribacteraceae, Chlorobi) were identified from the marine sediment-derived enrichments. Scanning electron microscopy (SEM) and catalyzed reporter deposition fluorescence in situ hybridization (CARD-FISH) images showed the ability of microorganisms to colonize and concurrently reduce magnetite likely stimulated by the observed methanogenic benzoate degradation. These findings explain the possible contribution of organoclastic reduction of iron oxides to the elevated dissolved Fe2+ pool typically observed in methanic zones of rapidly accumulating coastal and continental margin sediments

ONESIDED APPROXIMATION BY ENTIRE FUNCTIONS

Abstract. Let f: R → R have an nth derivative of finite variation V f (n) and a locally absolutely continuous (n − 1)st derivative. Denote by E ± (f, δ)p the error of onesided approximation of f (from above and below, respectively) by entire functions of exponential type δ&gt; 0 in L p (R)–norm. For 1 ≤ p ≤ ∞ we show the estimate with constants Cn&gt; 0. E ± (f, δ)p ≤ C 1−1/p n π 1/p Vf (n)δ −n − 1 p, 1

Entire Function Majorants

97 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2003.The third part contains some applications, among them Hilbert-type inequalities and inequalities of the form fx &le;cnsupy&isin; Rfn y for bounded functions f with a spectral gap at the origin.U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD