Scaling functions generating fractional Hilbert transforms of a wavelet function

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Author(s)

Abstract

It is well-known that an orthonormal scaling function generates an orthonormal wavelet function in the theory of multiresolution analysis. We consider two families of unitary operators. One is a family of extensions of the Hilbert transform called fractional Hilbert transforms. The other is a new family of operators which are a kind of modified translation operators. A fractional Hilbert transform of a given orthonormal wavelet (resp. scaling) function is also an orthonormal wavelet (resp. scaling) function, although a fractional Hilbert transform of a scaling function has bad localization in many cases. We show that a modified translation of a scaling function is also a scaling function, and it generates a fractional Hilbert transform of the corresponding wavelet function. We also show a good localization property of the modified translation operators. The modified translation operators act on the Meyer scaling functions as the ordinary translation operators. We give a class of scaling functions, on which the modified translation operators act as the ordinary translation operators.

Journal

  • Journal of the Mathematical Society of Japan

    Journal of the Mathematical Society of Japan 67(3), 1275-1294, 2015

    The Mathematical Society of Japan

Codes

  • NII Article ID (NAID)
    130005094184
  • NII NACSIS-CAT ID (NCID)
    AA0070177X
  • Text Lang
    ENG
  • ISSN
    0025-5645
  • NDL Article ID
    026603184
  • NDL Call No.
    Z53-A209
  • Data Source
    NDL  J-STAGE 
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