On Fourier transforms of wavelet packets

    Authors

    K. Ahmad; R. Kumar;L. Debnath

    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 < ( < omega > )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) < ( < 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)\ are continuous. (iii) \ < ( < omega > )over cap > (n)(xi)\(2) = Sigma (2r-1)(s=0)\ < ( < omega > )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)\ < ( < 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 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

    WOS:000171852000003

    ISSN

    0232-2064

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