TY - JOUR
T1 - The limits of refinable functions
AU - Strang, Gilbert
AU - Zhou, Ding-Xuan
PY - 2001
Y1 - 2001
N2 - A function $ is refinable (£ 6 5) if it is in the closed span of {ci(2i -k)}. This set S is not closed in Z/2(M), and \ve characterize its closure. A necessary and sufficient condition for a function to be refinable is presented without any information on the refinement mask. The Fourier transform of every / 6 5 \5 vanishes on a set of positive measure. As an example, we show that all functions with Fourier transform supported in TT, |TT]are the limits of refinable functions. The relation between a refinable function and its mask is studied, and nonuniqueness is proved. For inhomogeneous refinement equations we determine when a solution is refinable. This result is used to investigate refinable components of multiple refinable functions. Finally, we investigate fully refinable functions for which all translates (by any real number) are refinable. © 2001 American Mathematical Society.
AB - A function $ is refinable (£ 6 5) if it is in the closed span of {ci(2i -k)}. This set S is not closed in Z/2(M), and \ve characterize its closure. A necessary and sufficient condition for a function to be refinable is presented without any information on the refinement mask. The Fourier transform of every / 6 5 \5 vanishes on a set of positive measure. As an example, we show that all functions with Fourier transform supported in TT, |TT]are the limits of refinable functions. The relation between a refinable function and its mask is studied, and nonuniqueness is proved. For inhomogeneous refinement equations we determine when a solution is refinable. This result is used to investigate refinable components of multiple refinable functions. Finally, we investigate fully refinable functions for which all translates (by any real number) are refinable. © 2001 American Mathematical Society.
KW - Band-limited function
KW - Fourier transform
KW - Fully refinable function
KW - Inhomogeneous refinement equation
KW - Multiple refinable function
KW - Refinable function
KW - Refinement mask
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U2 - 10.1090/s0002-9947-01-02668-x
DO - 10.1090/s0002-9947-01-02668-x
M3 - RGC 21 - Publication in refereed journal
VL - 353
SP - 1971
EP - 1984
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
SN - 0002-9947
IS - 5
ER -