On Measure of Approximation by Means of Fourier Series

A many of approximation methods in C 2ß (Fej'er, de la Vall'ee Poussin etc.) may be generated via a certain function ' 2 C [0;1] with '(0) = 1, '(1) = 0. The function ' j (t) = cos(j \Gamma 1=2)ßt (j 2 N) generates the Rogosinski approximation method [N. K. Bari, "A Treatise on Trigonometric Series," I, II, Pergamon Press, 1964]. Our idea consists in representing ' by the orthogonal system ' j to extend results previously known for the Rogosinski method to arbitrary approximation methods. We illustrate this by proving two asymptotic estimates for the measure of approximation. 1. Introduction Let us consider the triangular -means U n (f; x) := a 0 2 + n X k=1 k (n)(a k cos kx + b k sin kx) (1) of the real Fourier series of a 2ß-periodical continuous function f 2 C 2ß .In the approximation theory [2,6] the following problem is set: find an asymptotic expansion for the quantity e(A; U n ) := sup f2A kf \Gamma U n fkC 2ß (2) called the measure of approximation of the class A ae..

Geometric properties of Gabor frames with a random window

In this paper we address various geometric properties of frames, such as spark, coherence, restricted isometry property, and frame order statistics. These properties play crucial role in various signal processing problems, including compressive sensing, phase retrieval, and quantization. We focus on a particular case of structured frames, namely on Gabor frames, where frame vectors are time and frequency shifts of a random window, and show that geometric properties of Gabor frames are close to optimum, which is usually demonstrated by Gaussian frames with independent vectors

Geometric properties of Gabor frames with a random window

In this paper we address various geometric properties of frames, such as spark, coherence, restricted isometry property, and frame order statistics. These properties play crucial role in various signal processing problems, including compressive sensing, phase retrieval, and quantization. We focus on a particular case of structured frames, namely on Gabor frames, where frame vectors are time and frequency shifts of a random window, and show that geometric properties of Gabor frames are close to optimum, which is usually demonstrated by Gaussian frames with independent vectors