Infinite-Dimensional Compressed Sensing and Function Interpolation

We introduce and analyse a framework for function interpolation using compressed sensing. This framework—which is based on weighted \(\ell ^1\) minimization—does not require a priori bounds on the expansion tail in either its implementation or its theoretical guarantees and in the absence of noise leads to genuinely interpolatory approximations. We also establish a new recovery guarantee for compressed sensing with weighted \(\ell ^1\) minimization based on this framework. This guarantee conveys several benefits. First, unlike existing results, it is sharp (up to constants and log factors) for large classes of functions regardless of the choice of weights. Second, by examining the measurement condition in the recovery guarantee, we are able to suggest a good overall strategy for selecting the weights. In particular, when applied to the important case of multivariate approximation with orthogonal polynomials, this weighting strategy leads to provably optimal estimates on the number of measurements required, whenever the support set of the significant coefficients is a so-called lower set. Finally, this guarantee can also be used to theoretically confirm the benefits of alternative weighting strategies where the weights are chosen based on prior support information. This provides a theoretical basis for a number of recent numerical studies showing the effectiveness of such approaches.     

Multivariate modified Fourier series and application to boundary value problems

In this paper we analyse the approximation-theoretic properties of modified Fourier series in Cartesian product domains with coefficients from both full and hyperbolic cross index sets. We show that the number of expansion coefficients may be reduced significantly whilst retaining comparable error estimates. In doing so we extend the univariate results of Iserles, Nørsett and S. Olver. We then demonstrate that these series can be used in the spectral-Galerkin approximation of second order Neumann boundary value problems, which offers some advantages over standard Chebyshev or Legendre polynomial discretizations.     

Divergent CH Insertion–Cyclization Cascades of N‐Allyl Ynamides

Gold carbene reactivity patterns were accessed by ynamide insertion into a C(sp³)? H bond. A substantial increase in molecular complexity occurred through the cascade polycyclization of N‐allyl ynamides to form fused nitrogen‐heterocycle scaffolds. Exquisite selectivity was observed despite several competing pathways in an efficient gold‐catalyzed synthesis of densely functionalized C(sp³)‐rich polycycles and a copper‐catalyzed synthesis of fused pyridine derivatives. The respective gold–keteniminium and ketenimine activation pathways have been explored through a structure–reactivity study, and isotopic labeling identified turnover‐limiting C? H bond‐cleavage in both processes.     

Stable reconstructions in Hilbert spaces and the resolution of the Gibbs phenomenon

We introduce a simple and efficient method to reconstruct an element of a Hilbert space in terms of an arbitrary finite collection of linearly independent reconstruction vectors, given a finite number of its samples with respect to any Riesz basis. As we establish, provided the dimension of the reconstruction space is chosen suitably in relation to the number of samples, this procedure can be implemented in a completely numerically stable manner. Moreover, the accuracy of the resulting approximation is determined solely by the choice of reconstruction basis, meaning that reconstruction vectors can be readily tailored to the particular problem at hand.An important example of this approach is the accurate recovery of a piecewise analytic function from its first few Fourier coefficients. Whilst the standard Fourier projection suffers from the Gibbs phenomenon, by reconstructing in a piecewise polynomial basis we obtain an approximation with root-exponential accuracy in terms of the number of Fourier samples and exponential accuracy in terms of the degree of the reconstruction. Numerical examples illustrate the advantage of this approach over other existing methods.     

Inhibition and enhancement of resonance as studied by 13C nuclear magnetic resonance spectroscopy

¹³C chemical shifts are reported for a series of 2-substituted 1,3-dimethylbenzenes: comparisons of these values with those for the corresponding monosubstituted benzenes reveal, in some cases, large differences in the para-carbon substituent chemical shifts, which are attributable to steric hindrance of resonance. The questions of steric enhancement of resonance, and methoxy group conformation in certain anisoles are also studied by the ¹³C NMR technique. Studies of selected 2-substituted fluorenes are also reported, and substituent chemical shifts at carbon-7 (traversing eight bonds) of greater than 2ppm are observed. These effects are consistently greater than those reported for the corresponding biphenyl compounds, and are associated with planarity-enforced enhancement of resonance.     

Aerosol direct fluorination: Direct synthesis of perfluorinated glyme ethers

Aerosol direct fluorination of 1,2-dimethoxyethane, bis(2-methoxyethyl)ether, triethyleneglycol dimethyl ether, and tetraethyleneglycol dimethyl ether produced the perfluorinated analogs in 36%, 22%, 30% and 15% isolated yields based on total input of hydrocarbon into the reactor. High effluent concentrations in excess of 80% were obtained in all cases indicating that low yields are primarily due to mechanical losses, not fragmentation or incomplete reaction. The clean reaction and the ability to produce high molecular weight fluorocarbons by direct fluorination suggests the applicability of aerosol fluorination to the production of fluorinated fluids and oils.     

Importance of entropy in the diastereoselectivity of 5-substituted 2-methyladamant-2-yl cations

The diastereofacial selectivity of 2-methyl-5-X-adamant-2-yl cations IX (X = CN, Cl, Br, CH3O, COOCH3, C6H5, CH3, and (CH3)3Sn) toward methanol has been investigated in the gas phase at 750 Torr and in the 40-120 degrees C temperature range and compared with that of IF (X = F) and ISi (X = (CH3)3Si) measured previously under similar conditions. Detailed analysis of the energy surface of the IMe (X = CH3) ion reveals that the activation barrier of its syn addition to methanol is significantly lower than that of the anti attack. In the 40-100 degrees C range, such a difference is strongly reduced by adverse entropic factors which are large enough to invert the IMe diastereoselectivity from syn to anti at T > 69 degrees C. The behavior of IMe diverges markedly from that of IF and ISi. Large adverse entropic factors account for the predominant syn diastereoselectivity observed in the reaction with IF (X = F), notwithstanding the anti enthalpy barrier is lower than the syn one. Adverse entropy plays a minor role in the reaction with ISi (X = (CH3)3Si) which instead exhibits a preferred anti diastereoselectivity governed by the activation enthalpies. Depending on the electronic properties of X, the kinetic behavior of the other IX ions obeys one of the above models. The gas-phase diastereoselectivity of IX ions responds to a subtle interplay between the sigma-hyperconjugative/electrostatic effects of the X substituent and the activation entropy terms. sigma-Hyperconjugation/field effects determine the pyramidal structure and the relative stability of the syn and anti conformers of IX as well as the relative stability of their addition transition structures and their position along the reaction coordinate. The diastereoselectivity of IX in the gas phase is compared with that measured in solution and with theoretical predictions.     

Substituent effects on fluorine - 19 chemical shifts in benzobicyclo(2.2.2)octen-1-yl fluorides

Inverse ¹⁹F substituent chemical shifts (SCS) of $\text{m}$- and $\text{p}$-substituted benzylfluorides are shown to have their origin in hyperconjugation involving the CF δ - bond.