Multiresolution analysis arising from Coalescence Hidden-variable Fractal Interpolation Functions (CHFIFs) is developed. The availability of a larger set of free variables and constrained variables with CHFIF in multiresolution analysis based on CHFIFs provides more control in reconstruction of functions in than that provided by multiresolution analysis based only on Affine Fractal Interpolation Functions (AFIFs). Our approach consists of introduction of the vector space of CHFIFs, determination of its dimension and construction of Riesz bases of vector subspaces , consisting of certain CHFIFs in . 1. Introduction The theory of multiresolution analysis provides a powerful method to construct wavelets having far reaching applications in analyzing signals and images [1, 2]. They permit efficient representation of functions at multiple levels of detail; that is, a function , the space of real valued functions satisfying , could be written as limit of successive approximations, each of which is smoothed version of . The multiresolution analysis was first introduced by Mallat [3] and Meyer [4] using a single function. The multiresolution analysis based upon several functions was developed in [5–7]. In [8], multiresolution analysis of was generated from certain classes of Affine Fractal Interpolation Functions (AFIFs). Such results were then generalized to several dimensions in [9, 10]. In [11], orthonormal basis for the vector space of AFIFs was explicitly constructed. A few years later, Donovan et al. [12] constructed orthogonal compactly supported continuous wavelets using multiresolution analysis arising from AFIFs. Bouboulis [13] generated multiresolution analysis of using AFIF on and constructed multiwavelets which are orthonormal but discontinuous. The interrelations among AFIFs, multiresolution analysis and wavelets are treated in [14]. In [15], multiresolution analysis is developed for a Hilbert space constructed from Hausdorff measures , on and, in particuluar, on a Cantor set using linear contraction. This development was later improved in [16] by employing a nonlinear fractal system to construct wavelets with Fourier basis with respect to some fractal measure. The details on implementation of recent work on multiresolution analysis with AFIF bases can be found in [17]. It is desirable [18] that the wavelet function should reflect the features present in the original function but AFIF based wavelets generally cannot exhibit satisfactorily the features of functions simulating natural objects or outcome of scientific experiments that are partly
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