Publication

Abstract : The Franklin wavelet is constructed using the multiresolution analysis (MRA) generated from a scaling function ϕ∈L2(R) that is continuous on R , linear on [2k,2k+1] and [2k+1,2k+2] for every k∈Z . For N=2,3,4 and α∈(0,1N−1) , it is shown that if a function ϕ∈L2(R) is continuous on R , linear on [Nk+Nlα,Nk+N(l+1)α] and [Nk+N(N−1)α,Nk+N] , for l=0,1,…,N−2 , and generates MRA with dilation factor N , then α=1N . Conversely, for α=1N,N=2,3,4 , it is shown that there exists a ϕ∈L2(R) , as satisfying the above conditions, that generates MRA with dilation factor N . The frame MRA (FMRA) is useful in signal processing, since the perfect reconstruction filter banks associated with FMRA can be narrow-band. So it is natural to ask, whether the above results can be extended for the case of FMRA. In this paper, for N=2 , we prove that if ϕ generates FMRA with dilation factor 2 , then α=12 . For N=3,4 , we prove similar results when α∈(0,1N−1)∖{12N} . In addition, for α=1N we prove that there exists a function ϕ∈L2(R), as satisfying the above conditions, that generates FMRA. Also, we construct tight wavelet frame and wavelet frame for such scaling functions.