Publication

Abstract : The Franklin wavelet (piecewise linear wavelet) is a wavelet function ψ∈L2(R) with dilation factor 2 which is continuous on R , linear on [k,k+12] and [k+12,k+1] for all k∈Z ; which is constructed from a multiresolution analysis(MRA) with a scaling function ϕ∈L2(R) which is continuous on R , linear on [2k,2k+1] and [2k+1,2k+2] for every k∈Z . Wavelets with composite dilation factor have the ability to produce long and narrow window functions, with varied orientations, well-suited to applications of image processing; also it is more effective and flexible for the construction of multiscale image representation. This motivates us to consider the following problem to construct some composite dilation Franklin wavelets. Given a natural number N≥2 , we ask, is there a continuous ϕ∈L2(R) that is linear on [Nk+Nlα,Nk+N(l+1)α] and [Nk+N(N−1)α,Nk+N] , for l=0,1,…,N−2 , α∈(0,1N−1) and for all k∈Z , which generates an MRA with dilation factor N to construct an orthonormal N - wavelet, i.e. a wavelet with dilation factor N ? For N=2 and α∈(0,1) we have proved that ϕ∈L2(R) generates an MRA with dilation factor 2 only if α=12 . Next for N=3,4 and α∈(0,1N−1)∖{12N} , we have proved that, if ϕ∈L2(R) generates an MRA with dilation factor N , then α=1N . Also, for α=1N,N=2,3,4 , we have proved existence of a continuous ϕ∈L2(R) which generates an MRA with dilation factor N . We have also constructed the corresponding orthonormal N -wavelet.