A study on the mask of interpolatory symmetric subdivision schemes

Ko, Kwan Pyo; Lee, Byung-Gook; Yoon, Gangjoon

doi:10.1016/j.amc.2006.08.089

Cited by 16 publications

(6 citation statements)

References 11 publications

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“…For binary schemes (m = 2) this was proven by Cavaretta et al [1] and Dyn [7], and a proof for general arity can be found in [12] and [18], for example. An alternative form of condition (4) is…”

Section: Algebraic Toolsmentioning

confidence: 81%

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Polynomial reproduction for univariate subdivision schemes of any arity

Conti

Hormann

2011

Journal of Approximation Theory

105

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In this paper, we study the ability of convergent subdivision schemes to reproduce polynomials in the sense that for initial data, which is sampled from some polynomial function, the scheme yields the same polynomial in the limit. This property is desirable because the reproduction of polynomials up to some degree d implies that a scheme has approximation order d + 1. We first show that any convergent, linear, uniform, and stationary subdivision scheme reproduces linear functions with respect to an appropriately chosen parameterization. We then present a simple algebraic condition for polynomial reproduction of higher order. All results are given for subdivision schemes of any arity m ≥ 2 and we use them to derive a unified definition of general m-ary pseudo-splines. Our framework also covers non-symmetric schemes and we give an example where the smoothness of the limit functions can be increased by giving up symmetry.

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Section: Algebraic Toolsmentioning

confidence: 81%

“…Now we use the induction hypothesis to manipulate the left-hand side of condition (18) and get for any…”

Section: Polynomial Reproductionmentioning

confidence: 99%

Polynomial reproduction for univariate subdivision schemes of any arity

Conti

Hormann

2011

Journal of Approximation Theory

105

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“…a j z j , z ∈ C \ {0} (1.5) as τ a = A (1) m (see, e.g., [5]) where A (z) denotes the first derivative of A(z). In particular, in [5] it is proven that, in case of a subdivision scheme with odd-symmetry (see Definition 2.2), τ a ∈ Z; conversely, in case of even-symmetry (see Definition 2.3), τ a ∈ Z 2 \ Z. Symmetric, compactly supported interpolating m-refinable functions that we can find in the literature always fulfill equation (1.4) in the case τ a ∈ Z, i.e., are associated with an odd-symmetric subdivision mask a satisfying (1.3) for an even integer M (see, e.g., [1,2,8,11,15,16,18,23,24]). In addition, for all of them the odd-symmetric subdivision mask a is such that…”

Section: Introduction and Purpose Of The Workmentioning

confidence: 99%

Interpolating m-refinable functions with compact support: The second generation class

Romani

2019

Applied Mathematics and Computation

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We present an algorithm for the construction of a new class of compactly supported interpolating refinable functions that we call the second generation class since, contrary to the existing class, is associated to subdivision schemes with an even-symmetric mask that does not contain the submask {0..., 0, 1, 0, ...0}.As application examples of the proposed algorithm we present interpolating 4-refinable functions that are generated by parameter-dependent, even-symmetric quaternary schemes never considered in the literature so far.

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“…Brief review of higher arity schemes having even-point complexity is presented below. Ko et al [11] introduced even point binary and ternary interpolating symmet ric subdivision schemes. Mustafa and Khan [13] introduced a new 4-point C3 qua ternary approximating subdivision scheme.…”

Section: Introductionmentioning

confidence: 99%

Generalized and Unified Families of Interpolating Subdivision Schemes

Mustafa¹

2014

NMTMA

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We present generalized and unified families of (2n)-point and (2n -1)point p-ary interpolating subdivision schemes originated from Lagrange polynomial for any integers n > 2 and p > 3. Almost all existing even-point and odd-point interpolating schemes of lower and higher arity belong to this family of schemes. We also present tensor product version of generalized and unified families of schemes. M oreover error bounds betw een limit curves and control polygons of schemes are also calculated. It has been observed th at error bounds decrease w hen complexity of the scheme decrease and vice versa. Furtherm ore, error bounds decrease w ith the increase of arity of the schemes. We also observe that in general the continuity of interpolating scheme do not increase by increasing complexity and arity of the scheme.

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A study on the mask of interpolatory symmetric subdivision schemes

Cited by 16 publications

References 11 publications

Polynomial reproduction for univariate subdivision schemes of any arity

Polynomial reproduction for univariate subdivision schemes of any arity

Interpolating m-refinable functions with compact support: The second generation class

Generalized and Unified Families of Interpolating Subdivision Schemes

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