Exponential polynomial reproducing property of non-stationary symmetric subdivision schemes and normalized exponential B-splines

Journal of Mathematical Analysis and Applications

2016

Pseudo-splines are a rich family of functions that allows the user to meet various demands for balancing polynomial reproduction (i.e., approximation power), regularity and support size. Such a family includes, as special members, B-spline functions, universally known for their usefulness in different fields of application. When replacing polynomial reproduction by exponential polynomial reproduction, a new family of functions is obtained. This new family is here constructed and called the family of exponential pseudo-splines. It is the nonstationary counterpart of (polynomial) pseudo-splines and includes exponential B-splines as a special subclass. In this work we provide a computational strategy for deriving the explicit expression of the Laurent polynomial sequence that identifies the family of exponential pseudo-spline nonstationary subdivision schemes. For this family we study its symmetry properties and perform its convergence and regularity analysis. Finally, we also show that the family of primal exponential pseudo-splines fills in the gap between exponential B-splines and interpolatory cardinal functions. This extends the analogous property of primal pseudo-spline stationary subdivision schemes.

“…Note that similar results concerning the normalization of exponential B-spline symbols are also given in [32]. Two special situations are considered in the next result.…”

Section: The Solution Of This System In the Unknowns P And K (K)mentioning

confidence: 53%

“…The above proposition proves the equivalence between the conditions for exponential polynomial reproduction given in [17] and in [32,33] when p = 0 or p = − 1 2 .…”

Section: Remark 36 As a By-product Of The Proof Of Proposition 35mentioning

confidence: 53%

Section: Non-stationary Subdivision Schemes and Exponential Polynomiamentioning

confidence: 99%

See 1 more Smart Citation

Exponential pseudo-splines: Looking beyond exponential B-splines

Conti

Gemignani

Journal of Mathematical Analysis and Applications

2016

“…As an example, we can think of the fact that stationary subdivision schemes are not capable of representing conic sections or, in general, exponential polynomials. On the contrary, nonstationary schemes can also generate exponential polynomials or exponential B-splines, that is piecewisely defined exponential polynomials [1,6,11,21,26,30,33,34,37,38]. Reproduction of piecewise exponential polynomials is important in several applications, e.g., in biomedical imaging, in geometric design and in isogeometric analysis.…”

Section: Introductionmentioning

confidence: 99%

Approximation order and approximate sum rules in subdivision

Conti

Journal of Approximation Theory

Yoon

2016

Self Cite

Several properties of stationary subdivision schemes are nowadays well understood. In particular, it is known that the polynomial generation and reproduction capability of a stationary subdivision scheme is strongly connected with sum rules, its convergence, smoothness and approximation order. The aim of this paper is to show that, in the non-stationary case, exponential polynomials and approximate sum rules play an analogous role of polynomials and sum rules in the stationary case. Indeed, in the non-stationary univariate case we are able to show the following important facts: i) reproduction of N exponential polynomials implies approximate sum rules of order N ; ii) generation of N exponential polynomials implies approximate sum rules of order N , under the additional assumption of asymptotical similarity and reproduction of one exponential polynomial; iii) reproduction of an N -dimensional space of exponential polynomials and asymptotical similarity imply approximation order N ; iv) the sequence of basic limit functions of a nonstationary scheme reproducing one exponential polynomial converges uniformly to the basic limit function of the asymptotically similar stationary scheme.

“…, x n−1 , e tx , e −tx } is proven. Moreover, we notice that We conclude by observing that the proposed non-stationary extension of the Lane-Riesenfeld algorithm offers an alternative definition of the symbols of normalized exponential B-splines recently introduced in [15,16].…”

Section: Proofmentioning

confidence: 99%

Building blocks for designing arbitrarily smooth subdivision schemes with conic precision

Novara

Journal of Computational and Applied Mathematics

2015

Since subdivision schemes featured by high smoothness and conic precision are strongly required in many application contexts, in this work we define the building blocks to obtain new families of non-stationary subdivision schemes enjoying such properties. To this purpose, we firstly derive a non-stationary extension of the Lane-Riesenfeld algorithm, and we exploit the resulting class of schemes to design a non-stationary family of alternating primal/dual subdivision schemes, all featured by reproduction of {1, x, e tx , e −tx }, t ∈ [0, π) ∪ iR + . Then, we focus our attention on interpolatory subdivision schemes with conic precision, that can be obtained as a byproduct of the above classes. In particular, we present a novel construction of a family of non-stationary interpolatory 2n-point schemes which generalizes the well-known Dubuc-Deslauriers family in such a way the n-th (n ≥ 2) family member reproduces Π 2n−3 ∪ {e tx , e −tx }, t ∈ [0, π) ∪ iR + , and keeps the original smoothness of its stationary counterpart unchanged.