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• W1686067509 abstract "A finitely supported sequence $a$ that sums to $2$ defines a scaling operator $T_a f = sum _{kin mathbb Z} a(k)f(2 cdot -k)$ on functions $f,$ a transition operator $S_a v = sum _{kin mathbb Z} a(k) (2 cdot -k)$ on sequences $v,$ and a unique compactly supported scaling function $phi$ that satisfies $phi = T_a phi$ normalized with $widehat phi (0) = 1.$ It is shown that the eigenvalues of $T_a$ on the space of compactly supported square-integrable functions are a subset of the nonzero eigenvalues of the transition operator $S_a$ on the space of finitely supported sequences, and that the two sets of eigenvalues are equal if and only if the corresponding scaling function $phi$ is a uniform $B$-spline." @default.
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• W1686067509 date "2005-07-21" @default.
• W1686067509 modified "2023-09-24" @default.
• W1686067509 title "Eigenvalues of scaling operators and a characterization of $B$-splines" @default.
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• W1686067509 doi "https://doi.org/10.1090/s0002-9939-05-08092-5" @default.
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