Spline collocation for strongly elliptic equations on the torus

Summary We prove convergence and error estimates in Sobolev spaces for the collocation method with tensor product splines for strongly elliptic pseudodifferential equations on the torus. Examples of applications include elliptic partial differential equations with periodic boundary conditions but also the classical boundary integral operators of potential theory on torus-shaped domains in three or more dimensions. For odd-degree splines, we prove convergence of nodal collocation for any strongly elliptic operator. For even-degree splines and midpoint collocation, we find an additional condition for the convergence which is satisfied for the classical boundary integral operators. Our analysis is a generalization to higher dimensions of the corresponding analysis of Arnold and Wendland [4].     

Asymptotic Distribution of Nodes for Near-Optimal Polynomial Interpolation on Certain Curves in R 2

Let E\subset \Bbb R ^s be compact and let d _n ^E denote the dimension of the space of polynomials of degree at most n in s variables restricted to E . We introduce the notion of an asymptotic interpolation measure (AIM). Such a measure, if it exists , describes the asymptotic behavior of any scheme τ _n ={ \bf x _k,n } _k=1 ^dnE , n=1,2,\ldots , of nodes for multivariate polynomial interpolation for which the norms of the corresponding interpolation operators do not grow geometrically large with n . We demonstrate the existence of AIMs for the finite union of compact subsets of certain algebraic curves in R ² . It turns out that the theory of logarithmic potentials with external fields plays a useful role in the investigation. Furthermore, for the sets mentioned above, we give a computationally simple construction for ``good' interpolation schemes. November 9, 2000. Date revised: August 4, 2001. Date accepted: September 14, 2001.     

Cardinal hermite interpolation with box splines II

Summary In the present work we extent the results in [RS] on CHIP, i.e. Cardinal Hermite Interpolation by the span of translates of directional derivatives of a box spline. These directional derivatives are that ones which define the type of the Hermite Interpolation. We admit here several (linearly independent) directions with multiplicities instead of one direction as in [RS]. Under the same assumptions on the smoothness of the box spline and its defining matrixT we can prove as in [RS]: CHIP has a system of fundamental solutions which are inL L ² together with its directional derivatives mentioned above. Moreover, for data sequences inl ^p ( ^d ), 1p2, there is a spline function inL ^p, 1/p+1/p=1, which solves CHIP.Research supported in part by NSERC Canada under Grant # A7687. This research was completed while this author was supported by a grant from the Deutscher Akademischer Austauschdienst     

Ion pair generation rates near planar3H and63Ni electron capture detector foils

The ion-pair generation rate (ionization topography) in plasmas from⁶³Ni and particularly Ti³H₄ foils, as used in electron capture detectors, was measured at room temperature using large, parallel plates of low backscattering ability in nitrogen gas of varying density. For one atmosphere pressure, the fall-off of ion pair formation as calculated from the exponential region equalsN ₀·e ^–0.19d for⁶³Ni andN ₀·e ^–1.4d for³H (whereN ₀ is the initial ionization rate immediately adjacent to the foil andd is the distance from the foil in mm). The experimentally measured half ranges (distances from the foil within which 50% of all possible ion pairs are created) are 2.7 mm for⁶³Ni and 0.27 mm for³H. The half ranges calculated from the exponential region where there is less interference from electron backscattering, are 3.7 and 0.5 mm, respectively. The latter values are considered closer to the true, unimpeded ionization topography near planar⁶³Ni and³H foils.Material taken from doctoral thesis     

On constrained multivariate splines and their approximations

Summary The variational formulation of multivariate spline functions is generalized to include cases where the function has to satisfy inequality constraints such as positivity and convexity. Condition for existence and uniqueness of a solution is given. Approximation to the solution can be obtained by solving the variational problem in a finite dimensional subspace. Conditions for convergence and error estimates of the approximations are presented, both for interpolation problems and smoothing problems. The general theory is illustrated by specific examples including the volume-matching problem and the one-sided thin plate spline.This research is partially supported by the U.S. Army Contract No. DAAG 29-77-G-0207, and by NSF Grant No. MCS-8101836Part of this paper is based on Chapters 2 and 3 of the author's Ph. D. thesis. The author would like to express his sincere thanks to Professor Grance Wahba and to two referees for many helpful comments     

~(14)N ENDOR Study on Bis(2-Hydroxylphenyl Ketoxime)-~(63)Cu(Ⅱ) Complex

The ENDOR spectrum of ~(14)N-~(63)Cu-HAP complex in DMSO/EtOH (5:1) freezing solution at 20 K has been studied using orientational selective method in this paper. The anisotropic superhyperfine coupling tensor and qusdrupole coupling tensor of ligand ~(14)N nucleus, and the superhyperfine coupling tensor of various ~1H nuclei have been measured precisely. Comparing the superhyperfine coupling tensor of ~(14)N-nucleus with previous work shows that the analytical method of spectrum is reasonable and the data are reliable in our previous work.     

Application of solvent extraction and synchronous spectrofluorimetry to the determination of the formation constant for a 1:1 complex of carbazole withβ-cyclodextrin in water-dimethyl sulfoxide medium

Extraction of carbazole in heptane was performed at 25±1°C with an aqueous dimethyl sulfoxide (DMSO) medium containing 63p0g110850/xxlarge946.gif" alt="beta" align="MIDDLE" BORDER="0">-cyclodextrin (63p0g110850/xxlarge946.gif" alt="beta" align="MIDDLE" BORDER="0">CD) at consecutive concentrations in the range of 0–10 mM. The fluorescence intensity of carbazole remaining in the heptane phase was measured by synchronous scanning fluorimetry. The apparent formation constant (K _f) for a 1:1 carbazole: 63p0g110850/xxlarge946.gif" alt="beta" align="MIDDLE" BORDER="0">CD inclusion complex in water-DMSO medium was determined by using a linear plot of the distribution ratio calculated from the fluorescence intensities vs. the 63p0g110850/xxlarge946.gif" alt="beta" align="MIDDLE" BORDER="0">-CD concentration. The values thus obtained ranged from 477 M^–1 in a 10% v/v DMSO medium to 12.1 M^–1 in a 60% v/v medium. Good linear relationships were observed between logK _f and the DMSO concentration ([DMSO]), and also between logK _f and the logarithm of the distribution coefficient (K _d) for carbazole. The formation constant in 100% water was estimated to be approximately 1.0×10³ M^–1 on the basis of the logK _f vs. [DMSO] and the logK _f vs. logK _d correlations.     

Quasi-interpolation with translates of a function having noncompact support

We establish a result related to a theorem of de Boor and Jia [1]. Their theorem, in turn, corrected and extended a result of Fix and Strang [5] concerning controlled approximation. In our result, the approximating functions are not required to have compact support, but satisfy instead conditions on their behavior at . Our theorem includes some recent results of Jackson [6] and is closely related to the work of Buhmann [2].Communicated by Carl de Boor     

Fourier analysis of the approximation power of principal shift-invariant spaces

The approximation order provided by a directed set {S _h } _h>0 of spaces, each spanned by thehZ ^d -translates of one function, is analyzed. The nearoptimal approximants of [R2] from eachs _h to the exponential functions are used to establish upper bounds on the approximation order. These approximants are also used on the Fourier transform domain to yield approximations for other smooth functions, and thereby provide lower bounds on the approximation order. As a special case, the classical Strang-Fix conditions are extended to bounded summable generating functions.The second part of the paper consists of a detailed account of various applications of these general results to spline and radial function theory. Emphasis is given to the case when the scale {s _h } is obtained froms ₁ by means other than dilation. This includes the derivation of spectral approximation orders associated with smooth positive definite generating functions.