#### Title

On Fourier transforms of wavelet packets

#### Abbreviated Journal Title

Z. Anal. ihre. Anwend.

#### Keywords

wavelet packets; multi-resolution analysis; Fourier transform; quadrature mirror filter; ORTHONORMAL BASES; MULTIRESOLUTION; Mathematics, Applied; Mathematics

#### Abstract

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.

#### Journal Title

Zeitschrift Fur Analysis Und Ihre Anwendungen

#### Volume

20

#### Issue/Number

3

#### Publication Date

1-1-2001

#### Document Type

Article

#### Language

English

#### First Page

579

#### Last Page

588

#### WOS Identifier

#### ISSN

0232-2064

#### Recommended Citation

"On Fourier transforms of wavelet packets" (2001). *Faculty Bibliography 2000s*. 2901.

http://stars.library.ucf.edu/facultybib2000/2901