On Fourier transforms of wavelet packets

This paper deals with the Fourier transform <(<omega>)over cap>(n), of wave let packets omega (n) is an element of L-2(R) relative to the scaling funct ion phi = omega (o). Included there are proofs of the following statements: (i) <(<omega>)over cap>n(0) = 0 for all n is an element of N. (ii) <(<omega>)over cap>(n) (4nk pi) = 0 for all k is an element of Z, n = 2(j) for some j is an element ofN(o), provided \<(<phi>)over cap>\,\m(o)\ a re continuous. (iii) \<(<omega>)over cap>(n)(xi)\(2) = Sigma (2r-1)(s=0)\<(<omega>)over ca p>(2rn+s)(2(r)xi)\(2) for r is an element of N. (iv) Sigma (infinity)(j=1) Sigma (2r-1)(s=0)Sigma (k is an element ofZ)\<(< omega>)over cap>(n)(2(j+r)(xi +2k pi))\(2) = 1 for a.a. xi is an element of R where r = 1,2,...,j. Moreover, several theorems including a result on quadrature mirror filter a re proved by using the Fourier transform of wavelet packets.