Regularity of generalized Daubechies wavelets reproducing exponential polynomials with real-valued parameters

Dyn, Nira; Kounchev, Ognyan; Levin, David; Render, Hermann

doi:10.1016/j.acha.2013.12.003

Cited by 20 publications

(49 citation statements)

References 33 publications

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“…Moreover, level-dependent subdivision schemes include Hermite schemes that do not only model curves and surfaces, but also their gradient fields (such schemes are again considered of interest both in geometric modelling and biological imaging, see, e.g., [8,9,11,27,34]). Additionally, non-stationary subdivision schemes are at the base of non-stationary wavelet and frame constructions that, being level adapted, are certainly more flexible [13,18,24,39]. Unfortunately, in practice, the use of subdivision is mostly restricted to the class of stationary subdivision schemes even though the non-stationary ones are equally simple to implement and highly intuitive in use: from an implementation point of view changing coefficients with the levels is not a crucial matter also in consideration of the fact that, in practice, only few subdivision iterations are performed.…”

Section: Motivationmentioning

confidence: 99%

Convergence and Normal Continuity Analysis of Nonstationary Subdivision Schemes Near Extraordinary Vertices and Faces

et al. 2019

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Convergence and normal continuity analysis of a bivariate non-stationary (leveldependent) subdivision scheme for 2-manifold meshes with arbitrary topology is still an open issue. Exploiting ideas from the theory of asymptotically equivalent subdivision schemes, in this paper we derive new sufficient conditions for establishing convergence and normal continuity of any rotationally symmetric, non-stationary, subdivision scheme near an extraordinary vertex/face.Keywords Non-stationary subdivision · Extraordinary vertex/face · Convergence · Normal continuity Mathematics Subject Classification (2010) 26A15 · 68U07 IntroductionThis paper provides a general procedure to check convergence of non-stationary (leveldependent) subdivision schemes in the neighborhood of an extraordinary vertex/face. It also gives sufficient conditions for the limit surface to be normal continuous at the limit point of an extraordinary vertex/face. To the best of our knowledge, the only contributions in this domain are the works in [7,21,26], where specific schemes are considered.

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Section: Motivationmentioning

confidence: 99%

Convergence and Normal Continuity Analysis of Nonstationary Subdivision Schemes Near Extraordinary Vertices and Faces

et al. 2019

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“…First of all, the infinite products a j k = ∞ r=1 ν j+r k and b j k = ∞ r=1 ν j+r k converges for any j ∈ N and k ∈ Z. Infinite products are considered convergent also in the case when they are equal to zero. Indeed, it follows from (6) and definition of ν j k that 1 ≥ ν j k ≥ ν j k ≥ 0. Therefore, log ν j k ≤ log ν j k , and we know that the series ∞ r=1 log ν j k is absolutely convergent or equal to −∞.…”

Section: Adjustment Of Localizationmentioning

confidence: 99%

“…It is well known that it is impossible to construct stationary wavelet basis satisfying these properties. Further, nonstationary wavelets are studied in [5,6,10,18]. Concerning to periodic case, first, periodic wavelets are generated by periodization of stationary wavelet functions.…”

Section: Introductionmentioning

confidence: 99%

On a connection between nonstationary and periodic wavelets

Lebedeva

2017

Journal of Mathematical Analysis and Applications

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We compare frameworks of nonstationary nonperiodic wavelets and periodic wavelets. We construct one system from another using periodization. There are infinitely many nonstationary systems corresponding to the same periodic wavelet. Under mild conditions on periodic scaling functions, among these nonstationary wavelet systems, we find a system such that its time-frequency localization is adjusted with an angular-frequency localization of an initial periodic wavelet system. Namely, we get the following equality lim j→∞ U C B (ψ P j ) = lim j→∞ U C H (ψ N j ), where U C B and U C H are the Breitenberger and the Heisenberg uncertainty constants, ψ P j ∈ L 2 (T) and ψ N j ∈ L 2 (R) are periodic and nonstationary wavelet functions respectively.

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“…The corresponding refinable function φ Λn := lim k→∞ S a (k) S a (k−1) · · · S a (1) δ is the limit function of a non-stationary subdivision scheme {S a (k) , k ≥ 1} reproducing exponential polynomials, i.e., solutions of the ODE of order n with constant coefficients and with spectrum Λ n . The interested reader can find more details on the construction and properties of these wavelets ψ Λn , n ≥ 2, in [33]. Next we would like to mention the following two properties of these masks {a (k) , k ≥ 1}:…”

Section: Exact Hölder Regularity Of Generalized Daubechies Waveletsmentioning

confidence: 99%

Regularity of non-stationary subdivision: a matrix approach

et al. 2016

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In this paper, we study scalar multivariate non-stationary subdivision schemes with integer dilation matrix M and present a unifying, general approach for checking their convergence and for determining their Hölder regularity (latter in the case ). The combination of the concepts of asymptotic similarity and approximate sum rules allows us to link stationary and non-stationary settings and to employ recent advances in methods for exact computation of the joint spectral radius. As an application, we prove a recent conjecture by Dyn et al. on the Hölder regularity of the generalized Daubechies wavelets. We illustrate our results with several examples.

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Regularity of generalized Daubechies wavelets reproducing exponential polynomials with real-valued parameters

Cited by 20 publications

References 33 publications

Convergence and Normal Continuity Analysis of Nonstationary Subdivision Schemes Near Extraordinary Vertices and Faces

Convergence and Normal Continuity Analysis of Nonstationary Subdivision Schemes Near Extraordinary Vertices and Faces

On a connection between nonstationary and periodic wavelets

Regularity of non-stationary subdivision: a matrix approach

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