On the adaptive spectral approximation of functions using redundant sets and frames

The approximation of smooth functions with a spectral basis typically leads to rapidly decaying coefficients, where the rate of decay depends on the smoothness of the function and vice versa. The optimal number of degrees of freedom in the approximation can be determined with relative ease by truncating the coefficients once a threshold is reached. Recent approximation schemes based on redundant sets and frames extend the applicability of spectral approximations to functions defined on irregular geometries and to certain nonsmooth functions. However, due to their inherent redundancy, the expansion coefficients in frame approximations do not necessarily decay even for very smooth functions. In this paper, we highlight this lack of equivalence between smoothness and coefficient decay, and we explore approaches to determine an optimal number of degrees of freedom for such redundant approximations.

On a localization property of wavelet coefficients for processes with stationary increments, and applications. I. Localization with respect to shift

We formulate a localization property of wavelet coefficients for processes with stationary increments, in the estimation problem associated with the processes. A general setting for the estimation is adopted and examples that fit this setting are given. An evaluation of wavelet coefficient decay with respect to shift k∈ℕ is explicitly derived (only the asymptotic behavior, for large k, was previously known). It is this evaluation that makes it possible to establish the localization property of the wavelet coefficients. In doing so, it turns out that the theory of positive-definite functions plays an important role. As applications, we show that, in the wavelet coefficient domain, estimators that use a simple moment method are nearly as good as maximum likelihood estimators. Moreover, even though the underlying process is long-range dependent and process domain estimates imply the validity of a noncentral limit theorem, for the wavelet coefficient domain estimates we always obtain a central limit theorem with a small prescribed error.

On a localization property of wavelet coefficients for processes with stationary increments, and applications. I. Localization with respect to shift

We formulate a localization property of wavelet coefficients for processes with stationary increments, in the estimation problem associated with the processes. A general setting for the estimation is adopted and examples that fit this setting are given. An evaluation of wavelet coefficient decay with respect to shift k∈ℕ is explicitly derived (only the asymptotic behavior, for large k, was previously known). It is this evaluation that makes it possible to establish the localization property of the wavelet coefficients. In doing so, it turns out that the theory of positive-definite functions plays an important role. As applications, we show that, in the wavelet coefficient domain, estimators that use a simple moment method are nearly as good as maximum likelihood estimators. Moreover, even though the underlying process is long-range dependent and process domain estimates imply the validity of a noncentral limit theorem, for the wavelet coefficient domain estimates we always obtain a central limit theorem with a small prescribed error.

Platinum Porphyrins as Phosphorescent Label for Time-resolved Microscopy

We investigated phosphorescent metalloporphyrins as potential labels for time-resolved microscopy. On the basis of spectroscopic analysis of their physicochemical properties (quantum yield, molar absorption coefficient, decay times) the best candidates were selected. Next, we synthesized antibody and avidin metalloporphyrin conjugates. The optimal F/P ratio with respect to quantum yield, decay time, and retention of biological activity of these immunoreagents was determined. The reagents were then evaluated by in situ hybridization and immunocytochemical procedures for demonstration of hapten-labeled DNA probes, membrane antigens (CD type), and 28S rRNA. All stained samples exhibited bright phosphorescence that could be selectively detected using time-resolved microscopy, especially when glucose/glucose oxidase was added to the embedding medium to deplete oxygen. Applications of time-resolved detection of phosphorescent porphyrins in strongly autofluorescent material (histological sections) are discussed.