The bestL 2-approximation by finite sums of functions with separable variables

Summary We consider the problem of the best approximation of a given functionh L ² (X × Y) by sums _k=1 ⁿ f _k f _k, with a prescribed numbern of products of arbitrary functionsf _k L ² (X) andg _k L ² (Y). As a co-product we develop a new proof of the Hilbert—Schmidt decomposition theorem for functions lying inL ² (X × Y).     

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.     

Remez-, Nikolskii-, and Markov-type inequalities for generalized nonnegative polynomials with restricted zeros

Sharp Remez-, Nikolskii-, and Markov-type inequalities are proved for functions of the form

A family of Hermite interpolants by bisection algorithms

A two point subdivision scheme with two parameters is proposed to draw curves corresponding to functions that satisfy Hermite conditions on [a, b]. We build two functionsf andf ¹ on dyadic numbers and for some values of the parameters,f is in ¹ withf ¹=f. Examples are provided which show how different the curves can be.     

A note on debye potentials for spherically gyrotropic media

The Debye potentials are generalized to the case of electromagnetic fields in spherically gyrotropic media. A medium is called spherically gyrotropic if it is locally gyrotropic with the distinguished axis having a radial direction determined by a central point. Expressions for electromagnetic fields in terms of the generalized potentials are presented and the system of differential equations for the potentials is derived. The results are summarized in the form of a theorem. Basic facts about the Debye potentials in isotropic media are recalled.     

Fabrication of MCM-41 mesoporous silica through the self-assembly supermolecule of β-CD and CTAB

Using the self-assembly β-cyclodextrin (β-CD) and cetyltrimethylammonium bromide (CTAB) as structure-directing agents, high-quality ordered MCM-41 silicas have been prepared. Small-angle X-ray diffraction (SXRD), N₂ adsorption-desorption and scanning electron microscope (SEM) techniques were used to characterize the calcined samples. Results showed that the pore structure of the resulting mesoporous silica belonged to the two-dimensional hexagonal structure (space group p6mm). The high-quality hexagonal structure was formed as n?1 (n denotes molar ratio of β-CD to CTAB). N₂ adsorption-desorption curves revealed type IV isotherms, H4 hysteresis loops, for all samples, and H1 hysteresis loops for samples at n=0, 0.1, 1 and 2, respectively. The pore size and the pore wall thickness of the samples increased with the increase in n values, respectively.     

应用理论计算研究MCM-22分子筛相邻酸性位上乙烯和苯的吸附

应用ONIOM计算方法研究了MCM-22分子筛超笼12元环上存在两个酸性位时的酸强度及其与骨架铝之间距离的关系,并研究了乙烯和苯分子吸附的规律.计算采用52T簇模型和B3LYP/6-31G**/MNDO方法.结果表明,存在两个酸性位且两个骨架铝之间间隔1个骨架硅时,酸强度比孤立的酸性位明显降低;当间隔的硅原子数增加时,酸强度呈上升趋势,间隔3个以上骨架硅时,其酸强度与孤立的酸性位几乎没有差别.对于乙烯的吸附,当两个骨架铝之间间隔1~4个骨架硅时,其吸附能几乎没有差别(31~35 kJ/mol);对于苯的吸附,当两个骨架铝之间间隔1个骨架硅时,其吸附能有所提高,因为两个桥羟基同时对苯分子产生氢键吸附作用.当两个骨架铝之间的距离增大时,苯的吸附能几乎相同(21~29 kJ/mol).若两个乙烯分子或苯分子同时吸附在双酸性位上,其吸附能与单个分子在孤立酸性位吸附时几乎没有差别.应用自然键轨道计算分析了吸附配合物的电子结构,进一步探明了乙烯和苯在分子筛酸性位上吸附的本质.     

载体MCM-41对Tb(aspirin)3phen发光性能的影响

在室温下,乙二胺环境中合成了高度有序的介孔材料MCM-41,并将经热处理的发光客体Tb(aspi-rin)3phen组装进其孔道,通过激发发射光谱对其光致发光性能进行了研究。结果表明,Tb(aspirin)3phen240～375 nm区间的宽激发峰归属于配体aspirin羰基n→π*跃迁、苯环π→π*跃迁,和phen的杂菲基团吸收,Tb3 的特征发射是由于Antenna效应引起的。相对于纯Tb(aspirin)3phen,Tb(aspirin)3phen-MCM-41B和Tb(aspirin)3phen/MCM-41A的激发谱带出现了明显的分裂,而Tb(aspirin)3phen-MCM-41A只在353nm处剩下了相对较窄的单峰。Tb(aspirin)3phen-MCM-41B,Tb(aspirin)3phen/MCM-41A和Tb(aspirin)3phen-MCM-41A的短波段激发峰依次减弱消失,长波段激发峰逐渐增强,而405 nm发射峰强度IL和544 nm发射峰强度ILn的比值I(I=IL/ILn)依次减小。MCM-41骨架与Tb(aspirin)3phen成键后,不同程度降低了配体aspirin和phen单重态S1和三重态T1能级,且对phen的影响大于aspirin。不同的MCM-41表面晶格场对配体能级的影响顺序为:MCM-41B外表面>MCM-41A外表面>MCM-41A内表面。I值可定性表示MCM-41表面晶格场对配体能级影响程度和MCM-41表面Tb(aspirin)3phen的含量。     

煅烧温度对Ba1.2Ca0.64SiO4：0.10Eu，0.06Mn粉体发光性能的影响

以MCM-41为硅源,在800~1 100℃下合成了Ba_1.2Ca_0.64SiO₄：0.10Eu,0.06Mn荧光粉,研究了煅烧温度对荧光粉发光性能的影响。实验结果表明,Eu²⁺的蓝绿光发射带的强度与宽度随煅烧温度的升高先增大后减小,在1 000℃时强度达到最大。最大发射峰位随温度的升高而发生红移,由800℃和900℃时的450 nm移至1 000℃和1 100℃时的480 nm。Mn²⁺的红光发射起源于Eu²⁺发射光的激发,其强度随温度的升高而增大。与Mn²⁺发生能量传递作用的Eu²⁺主要位于10配位的M（1）格位。