On Fourier transforms of wavelet packets

This paper deals with the Fourier transform \u3c ( \u3c omega \u3e )over cap \u3e (n), of wavelet packets omega (n) is an element of L-2(R) relative to the scaling function phi = omega (o). Included there are proofs of the following statements: (i) \u3c ( \u3c omega \u3e )over cap \u3e n(0) = 0 for all n is an element of N. (ii) \u3c ( \u3c omega \u3e )over cap \u3e (n) (4nk pi) = 0 for all k is an element of Z, n = 2(j) for some j is an element ofN(o), provided \ \u3c ( \u3c phi \u3e )over cap \u3e \,\m(o)\ are continuous. (iii) \ \u3c ( \u3c omega \u3e )over cap \u3e (n)(xi)\(2) = Sigma (2r-1)(s=0)\ \u3c ( \u3c omega \u3e )over cap \u3e (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)\ \u3c ( \u3c omega \u3e )over cap \u3e (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 are proved by using the Fourier transform of wavelet packets