Integral representation of Markov systems and the existence of adjoined functions for Haar spaces

AbstractLet A be a set of real numbers, and let Yn: = {y0,…, yn} be a C̆ebys̆ev system on A. Assume, moreover, that if inf A or sup A belongs to A, then it is a point of accumulation of A at which all yj are continuous. We find necessary and sufficient conditions for the existence of a function yn + 1 such that also {y0,…, yn, yn + 1} is a C̆ebys̆ev system on A. This theorem generalizes earlier results of Zielke and of the author. The proof is based on an integral representation of Markov systems that slightly extends a previous result of Zielke

On the Stability of Frames and Riesz Bases

AbstractThe first part of this paper supplements the recent work of Heil and Christensen on the stability of frames in Banach and Hilbert spaces. After obtaining a multivariate version of Kadec′s 1/4-theorem (which is used in the sequel), two of Christensen′s results, Chui and Shi′s Second Oversampling Theorem, and a variety of other results and techniques are applied to study the stability of multivariate exponential, wavelet, and Gabor frame and Riesz bases. Specific frame bounds and quantitative conditions of validity for mother wavelet and sampling perturbations are given