On Fourier transforms of wavelet packets
Abbreviated Journal Title
Z. Anal. ihre. Anwend.
wavelet packets; multi-resolution analysis; Fourier transform; quadrature mirror filter; ORTHONORMAL BASES; MULTIRESOLUTION; Mathematics, Applied; Mathematics
This paper deals with the Fourier transform <()over cap>(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) <()over cap>n(0) = 0 for all n is an element of N. (ii) <()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 \<()over cap>\,\m(o)\ are continuous. (iii) \<()over cap>(n)(xi)\(2) = Sigma (2r-1)(s=0)\<()over cap>(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)\<()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 are proved by using the Fourier transform of wavelet packets.
Zeitschrift Fur Analysis Und Ihre Anwendungen
"On Fourier transforms of wavelet packets" (2001). Faculty Bibliography 2000s. 2901.