The Characteristics Of Hilbert Frames And The Sums Of Frames

Frames for Hilbert spaces were introduced by Duffin and Schaeffer as a part of their research in non-harmonic Fourier series. Their work on frames was somewhat forgotten until 1986 when Daubechies, Grossmann and Meyer brought it all back to life during their fundamental work on wavelets. After this groundbreaking work, the theory of frames began to be more widely studied. The theory for frames and bases has developed very fast over the last 20 years. They are applied to signal processing, image processing, data compression and sampling fields.This paper is concerned with the characteristics of frames and Riesz basis and the conditions under which the sum of the frames is still a frame. The first chapter is introduction, In this part, we introduce origin of frame, progress and application, then outline the main content of this paper; In the second chapter, we give some fundamental definations about frames, Riesz basis and analysis operator; In the third chapter, we list the main results concerning the characteristics of Riesz basis and frames; Finally, we give the known theorems, then we give some conditions under which the theorem still stand.