Near Best Tree Approximation

Tree approximation is a form of nonlinear wavelet approximation that appears naturally in applications such as image compression and entropy encoding. The distinction between tree approximation and the more familiar n-term wavelet approximation is that the wavelets appearing in the approximant are required to align themselves in a certain connected tree structure. This makes their positions easy to encode. Previous work [4,6] has established upper bounds for the error of tree approximation for certain (Besov) classes of functions. This paper, in contrast, studies tree approximation of individual functions with the aim of characterizing those functions with a prescribed approximation error. We accomplish this in the case that the approximation error is measured in L ₂, or in the case p2, in the Besov spaces B _p ⁰(L _p ), which are close to (but not the same as) L _p . Our characterization of functions with a prescribed approximation order in these cases is given in terms of a certain maximal function applied to the wavelet coefficients.     

Composition-dependent photosensitivity in As-S glasses induced by bandgap light: structural origin by Raman scattering

Massive photoinduced short- and medium-range structural changes (photopolymerization) in As-S glasses are induced by near-bandgap light and studied by Raman scattering. Structural changes involve bond restructuring in sulfur-rich nanodomains of these nanoscale-phase-separated glasses. The spectral dependence of the photopolymerization effect demonstrates that various wavelengths can be used to optically change the structure of As-S glasses. The immense structural changes are relevant to recent findings about the role of bandgap light illumination for fabricating channel waveguides in noncrystalline arsenic sulfides.     

Effect of silver doping on the structure and phase separation of sulfur-rich As–S glasses: Raman and SEM studies

We report on the structural details and microphase separation of the bulk glasses Ag_x·(As₃₃S₆₇)_100-x for 0x25. Glass–glass phase separation occurs over a wide range of Ag content, i.e. 4x20. An off-resonant polarized Raman spectroscopic study has been carried out to elucidate structural aspects at the short- and medium-range structural order of the glasses. Analysis of Raman spectra revealed quantitative changes of the sulfur-rich microenvironments that reduce upon adding Ag. Scanning electron microscopy combined with X-rays microanalysis have been utilized to examine the type and extent of phase separation, and to provide quantitative details on the atomic concentrations in the Ag-poor and Ag-rich phases. It has been shown that at 7 at.% Ag the Ag-rich phase percolates through the structure; this effect can be associated with an ionic-to-superionic behavior of these glasses in accordance with similar studies on the stoichiometric arsenic sulfide glass; although the phase separation observed in the present glasses is qualitatively different.     

“Compactly” supported frames for spaces of distributions on the ball

Frames are constructed on the unit ball B ^d in ${\mathbb{R}^d}$ consisting of smooth functions with small shrinking supports. The new frames are designed so that they can be used for decomposition of weighted Triebel–Lizorkin and Besov spaces on B ^d with weight ${w_\mu(x):=(1-|x|^2)^{\mu-1/2}, \mu}$ half integer,?μ?≥ 0.     

Approximation of Distribution Spaces by Means of Kernel Operators

We investigate conditions on kernel operators in order to provide prescribed orders of approximation in the Triebel-Lizorkin spaces. Our approach is based on the study of the boundedness of integral kernel operators and extends the Strang-Fix theory, related to the approximation orders of principal shift-invariant spaces, to a wide variety of spaces.     

Decomposition of weighted Triebel-Lizorkin and Besov spaces on the ball

Weighted Triebel–Lizorkin and Besov spaces on the unitball B^d in ^d with weights w_µ(x)=(1–|x|²)^µ–1/2,µ0, are introduced and explored. A decomposition schemeis developed in terms of almost exponentially localized polynomialelements (needlets) {}, {} and it is shown that the membershipof a distribution to the weighted Triebel–Lizorkin orBesov spaces can be determined by the size of the needlet coefficients{f, } in appropriate sequence spaces.     

Approximation from shift-invariant spaces

We give sufficient conditions on a single function ? so that the principal shift-invariant space generated by ? provides a prescribed order of approximation inL _p (R ^d ), 1<p<∞, and inH _p (R ^d ), 0<p≤1. In particular, our conditions are given in terms of $\hat \varphi$ and are satisfied even when ? does not decay quickly at infinity.     

Atomic and Molecular Decomposition of Homogeneous Spaces of Distributions Associated to Non-negative Self-Adjoint Operators

We deal with homogeneous Besov and Triebel–Lizorkin spaces in the setting of a doubling metric measure space in the presence of a non-negative self-adjoint operator whose heat kernel has Gaussian localization and the Markov property. The class of almost diagonal operators on the associated sequence spaces is developed and it is shown that this class is an algebra. The boundedness of almost diagonal operators is utilized for establishing smooth molecular and atomic decompositions for the above homogeneous Besov and Triebel–Lizorkin spaces. Spectral multipliers for these spaces are established as well.