Generalized Hilbert Spaces And Multichannel Multiresolution Analysis For L～2(R)～n

Abstract. The concept of multiresolution analysis, formulated in 1988 by Mallat and Meyer, is very important in wavelet analysis, which provides a natural way to construct wavelet bases. The theory of multiresolution analysis has been developed and generalized by many authors. In this paper, we introduce the concepts of generalized Hilbert space and multichannel multiresolution analysis(MMRA) for L²(R)ⁿ, and discuss some important properties of them. This paper is organized as follows.In Chapter 1, we introduce and investigate generalized Hilbert space (?)^N×M which based on the concept of matrix-valued "inner-product". In light of Hilbert C^*-module, we show that a generalized Hilbert space is an orthogonally complemented Hilbert C^*-module.In Chapter 2, we introduce the concept of a multichannel multiresolution analysis. Associated with MMRA, we define scaling function and related wavelet are elements of L²(R)^n×n. Similar to the conventional MRA, we show that every MMRA give rise to an orthogonal wavelet. To construct wavelet from an MMRA, we need to construct bandpass filter from lowpass filter, by using operator-theoretic methods, we proved the existence of bandpass filter. Also, we will see that the component spaces in an MMRA form a multiresolution analysis of multiplicity for L²(R).In Chapter 3, we give some necessary conditions and sufficient conditions for construct orthogonal scaling functions from lowpass filter for finite-length matrix-valued filter. Finally, some examples are given.