On orthogonal wavelets with oversampling property

Considers the orthogonal scaling functions with the following general oversampling property: for a fixed integer J⩾0 and any f∈V₀, f(t)=Σ_nf(n/(2^J))φ(2^Jt-n). The authors call that an orthogonal scaling function φ(t) satisfying the above equation has the oversampling property with sampling rate 2^-J and denote all such scaling functions by S_J. Thus, S ₀ consists of all orthogonal scaling functions with the sampling property and S_J ⊂S_J+1 for J=0, 1, 2, …. The results in Walter (1993) also show that S₀≠S ₁, i.e., the space of the orthogonal scaling functions with the sampling property is a proper subspace of the one of the orthogonal scaling functions with the oversampling property. Let S_e and S_c denote all orthogonal scaling functions with exponential decay and compact support, respectively. The present authors prove that S₀∩S_e=S_J∩S_e and S ₀∩S_c=S_J∩S_c={the Haar function} for any J⩾O. In another words, if an orthogonal scaling function with exponential decay has the oversampling property then it has the sampling property, and if an orthogonal scaling function with compact support has the oversampling property then it must be the Haar function χ_0,1(t) which is 1 when 0⩽t⩽1 and 0 otherwise. These results imply that, weakening the sampling property to the oversampling proper6g on wavelets is not useful for finding more wavelet families with good decay properties