The Necessary Conditions For Frames

The concept of frame was introduced by R. J. Duffin and A. C. Schaeffer in 1952. Since 80's of the last century frames have been frequently applied in study of wavelets. For some popular families of functions in wavelet theory, people established some perfect characterization on frames, but for others sufficient conditions and necessary conditions were established seperatly. In 1990, I. Daubechies obtained a necessary condition for Gabor frames. In Chapter 1 of this paper we proved a set of new necessary conditions. These conditions are also sufficient for tight frames.In Chapter 2 of this paper we consider the necessary condition for affine frames. This is the main part of this paper. In 1990 I. Daubechies obtained the first result on necessary conditions for affine frames, and then in 1993 C. K. Chui and X. L. Shi obtained an improvement. Many books introduced their results as necessary conditions of affine frames. The main goal of Chapter 2 is to establish a set of necessary conditions. These conditions are also sufficient for tight frames.The aim of Chapter 3 is to discuss the sufficient condition for affine frames. The first result on sufficient condition of affine frames is obtained by I. Daubechies also. Recently X. L. Shi and Q. L. Shi proved a new criterion. In this chapter we established a different sufficient condition.In the fourth part we introduced so called a-adic summability method with applications to affine frames.