Trace Inequalities,Maximal Inequalities,and Weighted Fourier Transform Estimates

In the spirit of work of Kerman and Sawyer, a condition is given that is necessary and sufficient for the Fourier transform norm inequality $\Big(\int_{{\Bbb R}_d} \vert\hat{f}\vert^q d\mu\Big)^{1/q} \leq C\Big(\int_{{\Bbb R}_d} \vert f\vert^p v\Big)^{1/p}$ provided v is a radial weight for which v^?1/p is convexly decreasing and μ is a suitable measure. We also establish alternative conditions for such inequalities by proving corresponding trace type inequalities and maximal function inequalities that underlie the Fourier transform estimates. Our conditions are relatively simple to compute. Among applications we give extensions of a Sobolev restriction theorem.     

Constructive decomposition of functions of finite central mean oscillation

The space CMO of functions of finite central mean oscillation is an analogue of BMO where the condition that the sharp maximal function is bounded is replaced by the convergence of the sharp function at the origin. In this paper it is shown that each element of CMO is a singular integral image of an element of the Beurling space of functions whose Hardy-Littlewood maximal function converges at zero. This result is an analogue of Uchiyama's constructive decomposition of BMO in terms of singular integral images of bounded functions. The argument shows, in fact, that to each element of CMO one can construct a vector Calderón-Zygmund operator that maps that element into the proper subspace .

     

An Accurate In Vitro Model of the E. coli Envelope

Gram‐negative bacteria are an increasingly serious source of antibiotic‐resistant infections, partly owing to their characteristic protective envelope. This complex, 20 nm thick barrier includes a highly impermeable, asymmetric bilayer outer membrane (OM), which plays a pivotal role in resisting antibacterial chemotherapy. Nevertheless, the OM molecular structure and its dynamics are poorly understood because the structure is difficult to recreate or study in vitro. The successful formation and characterization of a fully asymmetric model envelope using Langmuir–Blodgett and Langmuir–Schaefer methods is now reported. Neutron reflectivity and isotopic labeling confirmed the expected structure and asymmetry and showed that experiments with antibacterial proteins reproduced published in vivo behavior. By closely recreating natural OM behavior, this model provides a much needed robust system for antibiotic development.     

On uncertainty bounds and growth estimates for fractional fourier transforms

Gelfand–Shilov spaces are spaces of entire functions defined in terms of a rate of growth in one direction and a concomitant rate of decay in an orthogonal direction. Gelfand and Shilov proved that the Fourier transform is an isomorphism among certain of these spaces. In this article we consider mapping properties of fractional Fourier transforms on Gelfand–Shilov spaces. Just as the Fourier transform corresponds to rotation through a right angle in the phase plane, fractional Fourier transforms correspond to rotations through intermediate angles. Therefore, the aim of fractional Fourier estimates is to set up a correspondence between growth properties in the complex plane versus decay properties in phase space.     

An Analogue of Slepian Vectors on Boolean Hypercubes

Journal of Fourier Analysis and Applications - Analogues of Slepian vectors are defined for finite-dimensional Boolean hypercubes. These vectors are the most concentrated in neighborhoods of the...     

Hardy's theorem and rotations

We prove an extension of Hardy's classical characterization of real Gaussians of the form , , to the case of complex Gaussians in which is a complex number with positive real part. Such functions represent rotations in the complex plane of real Gaussians. A condition on the rate of decay of analytic extensions of a function and its Fourier transform along some pair of lines in the complex plane is shown to imply that is a complex Gaussian.

     

The Definition of the Fourier Transform for Weighted Inequalities

Weighted norm inequalities for the Fourier transform have a long history, motivated in part by generalizations of the differentiation formula for Fourier transforms and by applications such as Kolmogorov′s prediction theory and restriction theory. Typically, a norm inequality, || ||L^q_μ ≤ ||f||L^p_v, is established on a subspace of L¹ contained in the weighted space L^p_v, where is the Fourier transform and μ and v are weights. The problem of defining the extension of on all of L^p_v is posed and solved here for several important cases. This definition sometimes results in the ordinary distributional Fourier transform, but there are other situations which are also analyzed. Precise necessary conditions and closure theorems are inextricably related to this program, and these results are established.     

Letter to the Editor: On the Numerical Evaluation of Bandpass Prolates

This letter provides a technique for numerical evaluation of certain eigenfunctions of the integral kernel operator corresponding to time truncation of a square-integrable function to a finite interval, followed by frequency limiting to frequencies in an annular band.     

Single peptide bonds exhibit poly(pro)II ("random coil") circular dichroism spectra

The far-UV circular dichroism spectra of a series of amino acid derivatives containing single peptide bonds have been measured. The N-acetyl-alanine displays a polyproline (PP) II-like spectrum, but alaninamide shows a very weak positive signal. Similarly Gly-Ala shows a PPII spectrum, but Ala-Gly does not. On heating, the spectrum shows a two-state transition also shown by long PPII polypeptides. Thus the characteristic PPII negative maximum at <200 nm results from the coupling of a peptide bond N-terminal to the chiral alpha-carbon, and therefore the simplest peptide bonds have a preferred conformation that defines the spectrum of disordered proteins of any size.