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Characterization of compactly supported refinable splines

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Abstract

We prove that a compactly supported spline functionφ of degree k satisfies the scaling equation \( \phi (x) = \sum _{n = 0}^N c(n)\phi (mx - n) \) for some integerm ≥ 2, if and only if \( \phi (x) = \sum _n p(n)B_k (x - n) \) wherep(n) are the coefficients of a polynomialP(z) such that the roots ofP(z)(z - 1)k+1 TM are mapped into themselves by the mappingzz m, andB k is the uniform B-spline of degreek. Furthermore, the shifts ofφ form a Riesz basis if and only ifP is a monomial.

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Lawton, W., Lee, S.L. & Shen, Z. Characterization of compactly supported refinable splines. Adv Comput Math 3, 137–145 (1995). https://doi.org/10.1007/BF03028364

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