Abstract. In this paper we wish to present a new class of tight frames on the sphere. These frames have excellent pointwise localization and approximation properties. These properties are based on pointwise localization of kernels arising in the spectral calculus for certain self-adjoint operators, and on a positive-weight quadrature formula for the sphere that the authors have recently developed. Improved bounds on the weights in this formula are another by-product of our analysis.

A discrete system of almost exponentially localized elements (needlets) on the n-dimensional unit sphere S n is constructed. It shown that the needlet system can be used for decomposition of Besov and Triebel-Lizorkin spaces on the sphere. As an application of Besov spaces on S n , a Jackson estimate for nonlinear m-term approximation from the needlet system is obtained.

Abstract. Classical and nonclassical Besov and Triebel-Lizorkin spaces with complete range of indices are developed in the general setting of Dirichlet space with a doubling measure and local scale-invariant Poincaré inequality. This leads to Heat kernel with small time Gaussian bounds and Hölder continuity, which play a central role in this article. Frames with band limited elements of sub-exponential space localization are developed, and frame and heat kernel characterizations of Besov and Triebel-Lizorkin spaces are established. This theory, in particular, allows to develop Besov and Triebel-Lizorkin spaces and their frame and heat kernel characterization in the context of Lie groups, Riemannian manifolds, and other settings.

Wavelet bases and frames consisting of band limited functions of nearly exponential localization on R d are a powerful tool in harmonic analysis by making various spaces of functions and distributions more accessible for study and utilization, and providing sparse representation of natural function spaces (e.g. Besov spaces) on R d . Such frames are also available on the sphere and in more general homogeneous spaces, on the interval and ball. The purpose of this article is to develop band limited well-localized frames in the general setting of Dirichlet spaces with doubling measure and a local scale-invariant Poincaré inequality which lead to heat kernels with small time Gaussian bounds and Hölder continuity. As an application of this construction, band limited frames are developed in the context of Lie groups or homogeneous spaces with polynomial volume growth, complete Riemannian manifolds with Ricci curvature bounded from below and satisfying the volume doubling property, and other settings. The new frames are used for decomposition of Besov spaces in this general setting.

High-speed videoendoscopy (HSV) captures the true intracycle vibratory behavior of the vocal folds, which allows for overcoming the limitations of videostroboscopy for more accurate objective quantification methods. However, the commercial HSV systems have not gained widespread clinical adoption because of remaining technical and methodological limitations and an associated lack of information regarding the validity, practicality, and clinical relevance of HSV. The purpose of this article is to summarize the practical, technological and methodological challenges we have faced, to delineate the advances we have made, and to share our current vision of the necessary steps towards developing HSV into a robust tool. This tool will provide further insights into the biomechanics of laryngeal sound production, as well as enable more accurate functional assessment of the pathophysiology of voice disorders leading to refinements in the diagnosis and management of vocal fold pathology. The original contributions of this paper are the descriptions of our color high-resolution HSV integration, the methods for facilitative playback and HSV dynamic segmentation, and the ongoing efforts for implementing HSV in phonomicrosurgery, as well as the analysis of the challenges and prospects for the clinical implementation of HSV, additionally supported by references to previously reported data.