The Convergence Of The Wavelet Series And Related Questions

There are two different kinds of wavelets. One has non-integrable scal-ing function such as wavelets of Shannon type; The other one has integrable scaling functions. A. Zayed ([21]) and X. L. Shi, W. Wang ([16])discussed the pointwise convergence of wavelet expansion for wavelets of Shannon type. S. E. Kelly, M. A. Kon and L. A. Raphael ([12] Established some criteria on pointwise convergence for the wavelets that have integrable scaling function-s. In this paper, we discuss the pointwise convergence of conjugate wavelet expansions for both these two kinds. The paper consists of four chapters.In the first chapter we introduce some basic concepts and main results.In the second chapter we present the Young theorem of conjugate Shannon wavelet approximation and establish a criterion for pointwise convergence of conjugate Shannon wavelet expansion of HBMV functions. This criterion is an improvement of Young theorem.In the third chapter we establish a criterion on almost everywhere con-vergence and LP convergence of conjugate Shannon wavelet expansion.In the fourth chapter we establish several criteria on pointwise convergence of conjugate wavelet expansion for the wavelets that have integrable scaling functions.