On orthogonal wavelets with oversampling property

Considers the orthogonal scaling functions with the following general oversampling property: for a fixed integer J/spl ges/0 and any f/spl isin/V/sub 0/, f(t)=/spl Sigma//sub n/f(n/(2/sup J/))/spl phi/(2/sup J/t-n). The authors call that an orthogonal scaling function /spl phi/(t) satisfying the above equation has the oversampling property with sampling rate 2/sup -J/ and denote all such scaling functions by S/sub J/. Thus, S/sub 0/ consists of all orthogonal scaling functions with the sampling property and S/sub J/ /spl sub/S/sub J+1/ for J=0, 1, 2, .... The results in Walter (1993) also show that S/sub 0//spl ne/S/sub 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/sub e/ and S/sub c/ denote all orthogonal scaling functions with exponential decay and compact support, respectively. The present authors prove that S/sub 0//spl cap/S/sub e/=S/sub J//spl cap/S/sub e/ and S/sub 0//spl cap/S/sub c/=S/sub J//spl cap/S/sub c/.<>