Convergence and summability of Gabor expansions at the Nyquist density

It is well known that Gabor expansions generated by a lattice of Nyquist density are numerically unstable, in the sense that they do not constitute frame decompositions. In this paper, we clarify exactly how bad such Gabor expansions are, we make it clear precisely where the edge is between enough and too little, and we find a remedy for their shortcomings in terms of a certain summability method. This is done through an investigation of somewhat more general sequences of points in the time-frequency plane than lattices (all of Nyquist density), which in a sense yields information about the uncertainty principle on a finer scale than allowed by traditional density considerations. An important role is played by certain Hilbert scales of function spaces, most notably by what we call the Schwartz scale and the Bargmann scale, and the intrinsically interesting fact that the Bargmann transform provides a bounded invertible mapping between these two scales. This permits us to turn the problems into interpolation problems in spaces of entire functions, which we are able to treat.     

On the stability of wavelet and Gabor frames (Riesz bases)

If the sequence of functions _{j, k} is a wavelet frame (Riesz basis) or Gabor frame (Riesz basis), we obtain its perturbation system _j,k which is still a frame (Riesz basis) under very mild conditions. For example, we do not need to know that the support of or is compact as in [14]. We also discuss the stability of irregular sampling problems. In order to arrive at some of our results, we set up a general multivariate version of Littlewood-Paley type inequality which was originally considered by Lemarié and Meyer [17], then by Chui and Shi [9], and Long [16].     

Spherical functions on harmonic extensions ofH-type groups

We consider the harmonic extension AN of an H-type group N with Lie algebra n = v + z, and [v, v] = z. We characterize the positive definite spherical functions on AN.     

Casimir effect in source theory III

A proper-energy method is used to find the energy of attraction between two dielectric slabs. Independent derivations of the attractive force are also presented.     

Influence of vapor transport equilibration on Raman spectra of Er:LiNbO3 crystals

Raman spectra of as-grown and vapor transport equilibration (VTE) treated Er:LiNbO₃ crystals, which have different cut orientations (X-cut and Z-cut), different Er-doping levels (Er:(0.2, 0.4 and 2.0 mol%)LiNbO₃) and different VTE durations (80, 120, 150 and 180 h), were recorded at room temperature in the wavenumber range 50-1000 cm⁻¹ by using backward scattering geometry. The spectra were attributed on the basis of their spectral features and the previous experimental work and the most recent theoretical progress in lattice dynamics on pure LiNbO₃. In comparison with the pure crystal the most remarkable effect of Er-doping on the Raman spectrum is observed for the E(TO₉) mode. It does not appear at 610 cm⁻¹ as the pure crystal, but locates at 633 cm⁻¹. In addition, the doping also results in the lowering of the Raman phonon frequency, the broadening of the Raman linewidth and the changes of the relative Raman intensity of some peaks. The VTE treatment results in the narrowing of the linewidth, the recovery of the lowered phonon frequency and the further changes of relative Raman intensity. The narrowing of Raman linewidth indicates that the VTE processing has brought these crystals closer to stoichiometric composition. The VTE treatment has induced the formation of a precipitate ErNbO₄ in the high-doped Er(2.0%):LiNbO₃ crystals whether X- or Z-cut. For these precipitated crystals, besides above linewidth and phonon frequency features, they also display more significant Raman intensity changes compared with those not precipitated crystals. In addition, a slight mixing between A₁(TO) and E(TO) spectra is also observed for these precipitated crystals. Above doping and VTE effects on Raman spectra were quantitatively or qualitatively correlated with the characteristics of the crystal structure and phonon vibrational system.     

Phase transition of Ta2NiO6 with the trirutile-type structure under high pressure and high temperature

High-pressure phase transition of Ta₂NiO₆ with the trirutile-type structure was investigated from the viewpoint of crystal chemistry. A new quenchable high-pressure phase was found in the pressure range higher than 7 GPa and 900°C. The high-pressure phase has an orthorhombic cell (a=4.797(1) Å, b=5.153(2) Å and c=14.85(1) Å and space group; Abm2), and it is more dense by 9.6% than the trirutile-structured phase. Infrared spectra of the trirutile-type phase and the high-pressure phase show that Ni²⁺ ions in the high-pressure phase are still in octahedral sites. The crystal structure of the high-pressure phase is considered as a cation-ordering trifluorite-type structure, which can be stabilized by a crystal field effect of Ni²⁺ ions.     

Electrical and optical properties of GexFexSe100−2x chalcogenide thin films

Optical absorption at room temperature and electrical conductivity at temperatures between 283 and 333 K of vacuum evaporated Ge_xFe_xSe_100−2x (0≤x≤15) amorphous thin films have been studied as a function of composition and film thickness. It was found that the optical absorption is due to indirect transition and the energy gap increases with increasing both Ge and Fe content; on the other hand, the width of the band tail exhibits the opposite behavior. The optical band gap E_opt was found to be almost thickness independent. The electrical conductivity show two types of conduction, at higher temperature the conduction is due to extended states, while the conduction at low temperature is due to variable range hopping in the localized states near Fermi level. Increasing Ge and Fe contents were found to decrease the localized state density N(E_F), electrical conductivity and increase the activation energy for conduction, which is nearly thickness independent. Variation of the atomic densities ρ, molar volume V, glass transition temperature T_g cohesive energy C.E and number of constraints N_Co with average coordination number Z was investigated. The relationship between the optical gap and chemical composition is discussed in terms of the cohesive energy C.E, average heat of atomization and coordination numbers.     

Numerical algorithms for uniform Airy-type asymptotic expansions

Airy-type asymptotic representations of a class of special functions are considered from a numerical point of view. It is well known that the evaluation of the coefficients of the asymptotic series near the transition point is a difficult problem. We discuss two methods for computing the asymptotic series. One method is based on expanding the coefficients of the asymptotic series in Maclaurin series. In the second method we consider auxiliary functions that can be computed more efficiently than the coefficients in the first method, and we do not need the tabulation of many coefficients. The methods are quite general, but the paper concentrates on Bessel functions, in particular on the differential equation of the Bessel functions, which has a turning point character when order and argument of the Bessel functions are equal.     

Optimal control and admissible relaxation for subdifferential evolution inclusions

In this paper we study the relaxation of optimal control problems monitored by subdifferential evolution inclusions. First under appropriate convexity conditions, we establish an existence result. Then we introduce the relaxed problem and show that it always has a solution under fairly general hypotheses on the data. Subsequently we examine when the relaxation is admissible. So we show that every relaxed trajectory can be approximated by extremal original ones (i.e. original trajectories generated by bang-bang controls) and that the values of the original and relaxed problems are equal. Some examples are also presented.