The Construction Of The L_p Space Multi-scale Analysis In The Compact Set And The Construction Of The Same Set Of Hermite Interpolating Wavelet

We solve the two problems which consist of " Multiresolution Analysis of L_p space on Compact Set" and "A Construction of Hermite Interpolating Wavelets".This article is composed of three chapters. In the first chapter, we have an introduction to the background of the two problems in this article, some concepts, notes, and the main results of this article.In the second chapter, we define Multiresolution Analysis (MRA) in function space L_p(Ω, μ) for p > 1, where Ω is a compact subset in Rⁿ and μ is a probability measure defined on Ω . We establish a general framework for construction of MRA in L_p(Ω, μ) and its conjugate space L_q(Ω, μ.) with p^-1 + q^-1 = 1, p , q > 1.We introduce the concept of a refinable set relative to a family of contractive mappings on a metric space.The notion of a refinable set parallels that of a refinable function,which is the basis of wavelet construction.The interpolation points we recursively generate from a refinable set by a set-theoretic multiresolution are analogous to multiresolution for functions used in wavelet construction.This set-theoretic multiresolution structure is named after set wavelets.The collection of set wavelets leads us to the construction of Hermite interpolation in L²[0,1] that has the desired multiscale structure in three chapter.