Pencho Petrushev

Localized Tight Frames on Spheres

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.

NONLINEAR APPROXIMATION AND THE SPACE BV (R2)

Given a function f 2 L2(Q), Q := [0, 1)2 and a real number t > 0, let U( f , t) := infg2BV (Q) kf gkL2(I) + t VQ (g), where the infimum is taken over all functions g 2 BV of bounded variation on I. This and related extremal problems arise in several areas of mathematics such as interpolation of operators and statistical estimation, as well as in digital image processing. Techniques for finding minimizers g for U( f , t) based on variational calculus and nonlinear partial differential equations have been put forward by several authors [DMS], [RO], [MS], [CL]. The main disadvantage of these approaches is that they are numerically intensive. On the other hand, it is well known that more elementary methods based on wavelet shrinkage solve related extremal problems, for example, the above problem with BV replaced by the Besov space B1(L1(I)) (see e.g. [CDLL]). However, since BV has no simple description in terms of wavelet coefficients, it is not clear that minimizers for U( f , t) can be realized in this way. We shall show in this paper that simple methods based on Haar thresholding provide near minimizers for U( f , t). Our analysis of this extremal problem brings forward many interesting relations between Haar decompositions and the space BV.

Clinical implementation of laryngeal high-speed videoendoscopy: Challenges and evolution

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.

Approximation by ridge functions and neural networks

We investigate the efficiency of approximation by linear combinations of ridge functions in the metric of L2 (Bd ) with Bd the unit ball in Rd . If Xn is an n-dimensional linear space of univariate functions in L2 (I), I=[-1,1], and

Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces

\Omega

New bases for Triebel-Lizorkin and Besov spaces

is a subset of the unit sphere Sd-1 in Rd of cardinality m, then the space Yn:={span}\{r({\bf x}\cdot\xi):r\in X_n,\omega\in\Omega\}

Nonlinear Wavelet Approximation in the Space C(Rd)

is a linear space of ridge functions of dimension

Nonlinear approximation from differentiable piecewise polynomials

\le mn

Multilevel preconditioning for partition of unity methods: some analytic concepts

. We show that if Xn provides order of approximation O(n-r ) for univariate functions with r derivatives in L2 (I), and

Multivariate n -term rational and piecewise polynomial approximation

\Omega