CHARACTERIZATIONS OF SCALING FUNCTIONS: CONTINUOUS SOLUTIONS∗

A dilation equation is a functional equation of the form f(t) = ∑ N k=0 ck f(2t − k), and any nonzero solution of such an equation is called a scaling function. Dilation equations play an important role in several fields, including interpolating subdivision schemes and wavelet theory. This paper obtains sharp bounds for the Hölder exponent of continuity of any continuous, compactly supported scaling function in terms of the joint spectral radius of two matrices determined by the coefficients {c0, . . . , cN}. The arguments lead directly to a characterization of all dilation equations that have continuous, compactly supported solutions.

[1]  M. Newman,et al.  Topics in Algebra , 1978 .

[2]  S. Dubuc.  Interpolation through an iterative scheme , 1986 .

[3]  I. Daubechies.  Orthonormal bases of compactly supported wavelets , 1988 .

[4]  C. Micchelli,et al.  Uniform refinement of curves , 1989 .

[5]  Gilles Deslauriers,et al.  Symmetric iterative interpolation processes , 1989 .

[6]  Gilbert Strang,et al.  Wavelets and Dilation Equations: A Brief Introduction , 1989, SIAM Rev..

[7]  H. Prautzsch,et al.  Refinement and subdivision for spaces of integer translates of a compactly supported function , 1989 .

[8]  S. Mallat.  Multiresolution approximations and wavelet orthonormal bases of L^2(R) , 1989 .

[9]  W. Lawton.  Tight frames of compactly supported affine wavelets , 1990 .

[10]  C. Micchelli,et al.  Stationary Subdivision , 1991 .

[11]  Nira Dyn,et al.  Analysis of uniform binary subdivision schemes for curve design , 1991 .

[12]  I. Daubechies,et al.  Two-scale difference equations I: existence and global regularity of solutions , 1991 .

[13]  I. Daubechies,et al.  Sets of Matrices All Infinite Products of Which Converge , 1992 .

[14]  Christopher Heil,et al.  The characterization of continuous, four-coefficient scaling functions and wavelets , 1992, IEEE Trans. Inf. Theory.

[15]  I. Daubechies,et al.  Two-scale difference equations II. local regularity, infinite products of matrices and fractals , 1992 .

[16]  T. Eirola.  Sobolev characterization of solutions of dilation equations , 1992 .

[17]  DaubechiesIngrid.  Orthonormal bases of compactly supported wavelets II , 1993 .

[18]  Gilbert Strang,et al.  CONTINUITY OF THE JOINT SPECTRAL RADIUS: APPLICATION TO WAVELETS , 1995 .

[19]  Yang Wang.  Two-scale dilation equations and the cascade algorithm , 1995 .