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  • 1.
    Gundy, Richard
    et al.
    Rutgers University.
    Jonsson, Adam
    Scaling Functions on R2 for Dilations with Determinant ±22010In: Applied and Computational Harmonic Analysis, ISSN 1063-5203, E-ISSN 1096-603X, Vol. 29, no 1, p. 49-62Article in journal (Refereed)
    Abstract [en]

    This paper gives necessary and sufficient conditions for a doubly periodic function p(ξ ), ξ ∈ R2, to be the squared modulus of a lowpass filter for a multiresolution analysis of L2(R2) with respect to an expanding matrix A of determinant ±2. By transferring the underlying spaces, R or R2, to a single binary sequence space, we are able to show that, when det(A) = 2, every scaling function on R2 corresponds to one on R, where the dilation is ±2. If det(A)=−2, this is no longer true. In this case, the lowpass filter for the stretched Haar function makes an unexpected appearance.

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