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A Necessary and Sufficient Proximity Condition for Smoothness Equivalence of Nonlinear Subdivision Schemes

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Abstract

In the recent literature on subdivision methods for approximation of manifold-valued data, a certain “proximity condition” comparing a nonlinear subdivision scheme to a linear subdivision scheme has proved to be a key analytic tool for analyzing regularity properties of the scheme. This proximity condition is now well known to be a sufficient condition for the nonlinear scheme to inherit the regularity of the corresponding linear scheme (this is called smoothness equivalence). Necessity, however, has remained an open problem. This paper introduces a smooth compatibility condition together with a new proximity condition (the differential proximity condition). The smooth compatibility condition makes precise the relation between nonlinear and linear subdivision schemes. It is shown that under the smooth compatibility condition, the differential proximity condition is both necessary and sufficient for smoothness equivalence. It is shown that the failure of the proximity condition corresponds to the presence of resonance terms in a certain discrete dynamical system derived from the nonlinear scheme. Such resonance terms are then shown to slow down the convergence rate relative to the convergence rate of the corresponding linear scheme. Finally, a super-convergence property of nonlinear subdivision schemes is used to conclude that the slowed decay causes a breakdown of smoothness. The proof of sufficiency relies on certain properties of the Taylor expansion of nonlinear subdivision schemes, which, in addition, explain why the differential proximity condition implies the proximity conditions that appear in previous work.

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Notes

  1. In [13, Definition3.5], Grohs gives a similar compatibility condition, which he calls a “differential proximity condition.”

  2. We assume that \(q_{\sigma }\) are \(C^{\infty }\), but our analysis only requires continuity of derivatives up to order \(k+1\), where k is the order of smoothness of \(S_\mathrm{lin}\).

  3. We wish to thank one of the referees for pointing us to this class of nonlinear subdivision rules.

  4. Other authors explore generalizations to settings where the domain space is multi-dimensional but for low order smoothness, see for example [9, 25].

  5. Note how this step would fail if we had an unknown constant \(C>1\) in front of the right-hand side of (3.17).

  6. This observation motivates the \(C^1\) proximity condition that first appeared in [22].

  7. Here, \(S_\mathrm{lin}\) reproduces \(\Pi _k\) means \(S_\mathrm{lin}(\Pi _k) \subset \Pi _k\), which is equivalent to the Fourier domain condition \(\widehat{a}^{(\ell )}(-1)=0\), \(0 \le \ell \le k\) [1, Lemma 3.1], or, equivalently, the time domain condition \(\sum _k a_{2k} \pi (k+1/2) = \sum _k a_{2k+1} \pi (k)\) for all \(\pi \in \Pi _k\). Here \(\widehat{a}^{(\ell )}(z) = \sum _k a_k z^{-k}\) is the symbol of the mask of \(S_\mathrm{lin}\).

  8. In this context, it means \(\Vert \Delta ^2 S_\mathrm{lin}^j x \Vert _\infty = O(2^{-j})\). In general, the Zgymund class [33] is the space of bounded functions which satisfy \(\sup _x | \Delta ^2_h f(x) | = O(h)\). In contrast, functions in \(C^{0,1}\) (=\(\mathrm{Lip} 1\)) satisfy \(\sup _x | \Delta _h f(x) | = O(h)\). It is well known (e.g., [17, 33]) that \(\Lambda _*\supsetneqq \mathrm{Lip} 1 \supsetneqq C^1\). Similarly, \(\Lambda ^{m+1}_*\) is the space of bounded functions with m-th derivatives in \(\Lambda _*\); we have \(C^{m}\supsetneqq \Lambda ^{m+1}_*\supsetneqq C^{m,1} \supsetneqq C^{m+1}\).

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Authors and Affiliations

  1. Department of Mathematics, University of Washington, Box 354350, Seattle, WA, 98195-4350, USA

    Tom Duchamp

  2. Department of Mathematics, East China University of Science and Technology, Shanghai, 200237, China

    Gang Xie

  3. Department of Mathematics, Drexel University, 3141 Chestnut Street, 206 Korman Center, Philadelphia, PA, 19104, USA

    Thomas Yu

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  1. Tom Duchamp
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Correspondence to Thomas Yu.

Additional information

Communicated by Arieh Iserles.

Tom Duchamp gratefully acknowledges the support and hospitality provided by the IMA during his visit from April to June 2011, when part of the work in this article was completed, as well as travel support through the PIMS CRG on Applied and Computational Harmonic Analysis. Gang Xie’s research was supported by the Fundamental Research Funds for the Central Universities and the National, Natural Science Foundation of China (No.11101146). Thomas Yu’s research was partially supported by the National Science Foundation grants DMS 0915068 and DMS 1115915, as well as a fellowship offered by the Louis and Bessie Stein family. The main result of this paper was first presented in the workshop “New trends in subdivision and related applications” held in the University of Milano-Bicocca, Italy in September 4–7, 2012. He thanks Dennis Yang, Georgi Medvedev, and Mark Levi for discussions on dynamical systems.

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Duchamp, T., Xie, G. & Yu, T. A Necessary and Sufficient Proximity Condition for Smoothness Equivalence of Nonlinear Subdivision Schemes. Found Comput Math 16, 1069–1114 (2016). https://doi.org/10.1007/s10208-015-9268-6

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