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The radial basis function (RBF) approximation is a rapidly developing field of mathematics. In the paper, we are concerned with the measurement of scalar physical quantities at nodes on sphere in the 3D Euclidean space and the spherical RBF interpolation of the data acquired. We employ a multiquadric as the radial basis function and the corresponding trend is a polynomial of degree 2 considered in Cartesian coordinates. Attention is paid to geodesic metrics that define the distance of two points on a sphere. The choice of a particular geodesic metric function is an important part of the construction of interpolation formula. We show the existence of an interpolation formula of the type considered. The approximation formulas of this type can be useful in the interpretation of measurements of various physical quantities. We present an example concerned with the sampling of anisotropy of magnetic susceptibility having extensive applications in geosciences and demonstrate the advantages and drawbacks of the formulas chosen, in particular the strong dependence of interpolation results on condition number of the matrix of the system considered and on round-off errors in general. [ABSTRACT FROM AUTHOR]

In this work, a general class of interpolation and smoothing natural exponential splines with respect to fourth order differential operators with two real parameters is considered. Some sufficient conditions for the associated matrix R to be a diagonally dominant matrix are given. Based on these, fast algorithms for computing the coefficients of this general class of exponential splines are developed. The obtained splines have C 2 continuity and are the minimum solution of the combination of interpolation and smoothing energy integral. The performances of the resulting splines in financial data from the S &P500 index are given. Numerical experiments show that the resulting splines have more freedom to adjust the shape and control the energy of the curves. Cross-validation and generalized cross-validation for determining an appropriate smoothing parameter are also given. [ABSTRACT FROM AUTHOR]

We describe a general approach for constructing a broad class of operators approximating high-dimensional curves based on geometric Hermite data. The geometric Hermite data consists of point samples and their associated tangent vectors of unit length. Extending the classical Hermite interpolation of functions, this geometric Hermite problem has become popular in recent years and has ignited a series of solutions in the 2D plane and 3D space. Here, we present a method for approximating curves, which is valid in any dimension. A basic building block of our approach is a Hermite average — a notion introduced in this paper. We provide an example of such an average and show, via an illustrative interpolating subdivision scheme, how the limits of the subdivision scheme inherit geometric properties of the average. Finally, we prove the convergence of this subdivision scheme, whose limit interpolates the geometric Hermite data and approximates the sampled curve. We conclude the paper with various numerical examples that elucidate the advantages of our approach. [ABSTRACT FROM AUTHOR]

In this work, a class of interpolation and smoothing exponential splines with respect to a sixth order differential operator with two parameters is constructed. All the square matrices involved in the construction are proved to be tridiagonal symmetric and diagonally dominant, which results in algorithms for computing this class of exponential splines. The obtained splines have C 4 continuity and are the minimum solution of the combination of interpolation and a generalized smoothing energy integral. The performances of the resulting splines in financial data from the S &P500 index and the effect of fitting multi-exponential decay data are given. Numerical experiments show that the resulting splines have more freedom to adjust the shape and control the energy of the curves and perform better than previous methods in fitting multi-exponential decay data. And cross validation and generalized cross validation for determining an appropriate smoothing parameter are also developed. [ABSTRACT FROM AUTHOR]

This paper deals with the construction of the Algebraic Trigonometric Pythagorean Hodograph (ATPH) cubic-like Hermite interpolant. A characterization of solutions according to the tangents at both ends and a global free shape parameter α is performed. Since this degree of freedom can be used for adjustments, we study how the curve evolves with respect to α. Several examples illustrating the construction process and a simple fitting method to determine the unique ATPH curve passing through a given point are proposed. [ABSTRACT FROM AUTHOR]

This paper introduces a novel technique to approximate a given continuous function f defined on a real compact interval by a new class of zipper α -fractal functions which contain a scaling vector and a binary vector or signature. For specific choices of scaling and signature vectors, the corresponding zipper fractal functions simultaneously interpolate and approximate f. When signature is zero, the proposed zipper fractal functions coincide with existing α -fractal functions. Hence, the zipper approximation proposed in this manuscript generalizes the existing fractal and classical approximations. Zipper fractal analogue of some elementary results in the classical approximation theory are obtained. Using convex optimization technique, we investigate the existence of optimal zipper fractal function for a given continuous function. The parameter identification problems for zipper α -fractal approximants are investigated. We derive sufficient conditions on the parameters of zipper α -fractal functions so that these functions preserve the positivity, monotonicity and convexity of the original function f. Also, we have studied the constructions of k-times continuously differentiable zipper α -fractal functions and one sided zipper fractal approximants for f. Numerical illustrations are provided to support the proposed theoretical results on zipper α -fractal functions. [ABSTRACT FROM AUTHOR]

The main purpose of this work is to provide an efficient scheme for constructing kernel-based reduced interpolation models. In the existing literature such problems are mainly addressed via the well-established knot insertion or knot removal schemes. Such iterative strategies are usually quite demanding from a computational point of view and our goal is to study an efficient implementation for data removal approaches, namely efficient reduced basis algorithm (ERBA). Focusing on kernel-based interpolation, the algorithm makes use of two iterative rules for removing data. The former, called ERBA-r, is based on classical residual evaluations. The latter, namely ERBA-p, is independent of the function values and relies on error bounds defined by the power function. In both cases, inspired by the so-called extended Rippa’s algorithm, our ERBA takes advantage of a fast implementation. [ABSTRACT FROM AUTHOR]

In this paper, we concern with the data interpolation by using extended cubic uniform B-splines with shape parameters. Two iterative formats, namely the progressive iterative approximation (PIA) and the weighted progressive iterative approximation (WPIA), are proposed to interpolate given data points. We study the optimal shape parameter and the optimal weight for the proposed methods by solving the eigenvalues of the collocation matrix. The optimal shape parameter can make the iterative methods not only have the fastest convergence speed but also have smallest initial interpolation error. Numerical experiments are given to illustrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

The data driven extrapolation requires the definition of a functional model depending on the available data and has the application scope of providing reliable predictions on the unknown dynamics. Since data might be scattered, we drive our attention towards kernel models that have the advantage of being meshfree. Precisely, the proposed numerical method makes use of the so-called Variably Scaled Kernels (VSKs), which are introduced to implement a feature augmentation-like strategy based on discrete data. Due to the possible uncertainty on the data and since we are interested in modelling the behaviour of the target functions, we seek for a regularized solution by ridge regression. Focusing on polyharmonic splines, we investigate their implementation in the VSK setting and we provide error bounds in Beppo–Levi spaces. The performances of the method are then tested on functions showing exponential or rational decay. Comparisons with Support Vector Regression (SVR) are also carried out and highlight that the proposed approach is effective, particularly since it does not require to train complex architecture constructions. [ABSTRACT FROM AUTHOR]

In this paper a special class of one-dimensional L-splines of order 4 is studied, which naturally appear in the computation of interpolation and smoothing with multivariate polysplines. Fast algorithms are provided for interpolation and smoothing with this class of L-splines, as well as a generalization of the Reinsch algorithm to this setting. The explicit description of all mathematical expressions permits a simple and direct numerical implementation. Applications are provided to financial data of the index S&P500, for the fast calculation of statistically interesting quantities, as cross validation (scores), generalized cross validation (scores) for finding the best smoothing parameter α , and the residual sum of squares. [ABSTRACT FROM AUTHOR]

The paper provides a general and simple approach for explicitly constructing multiscale radial kernels with high-order generalized Strang-Fix conditions from a given univariate generator. The resulting kernels are constructed by taking a linear functional to the scaled f -form of the generator with respect to the scale variable. Equivalent divided difference forms of the kernels are also derived; based on which, a pyramid-like algorithm for fast and stable computation of multiscale radial kernels is proposed. In addition, characterizations of the kernels in both the spatial and frequency domains are given, which show that the generalized Strang-Fix condition, the moment condition, and the condition of polynomial reproduction in the convolution sense are equivalent to each other. Hence, as a byproduct, the paper provides a unified view of these three classical concepts. These kernels can be used to construct quasi-interpolation with high approximation accuracy and construct convolution operators with high approximation orders, to name a few. As an example, we construct a quasi-interpolation scheme for irregularly spaced data and derived its error estimates and choices of scale parameters of multiscale radial kernels. Numerical results of approximating a bivariate Franke function using our quasi-interpolation are presented at the end of the paper. Both theoretical and numerical results show that quasi-interpolation with multiscale radial kernels satisfying high-order generalized Strang-Fix conditions usually provides high approximation orders. [ABSTRACT FROM AUTHOR]

Given discrete function values sampled at uniform centers, the iterated quasi-interpolation approach for approximating the m th derivative consists of two steps. The first step adopts m successive applications of the operator DQ (the quasi-interpolation operator Q first, and then the differentiation operator D) to get approximated values of the m th derivative at uniform centers. Then, by one further application of the quasi-interpolation operator Q to corresponding approximated derivative values gives the final approximation of the m th derivative. The most salient feature of the approach is that it approximates all derivatives with the same convergence rate. In addition, it is valid for a general multivariate function, compared with the existing iterated interpolation approaches that are only valid for periodic functions, so far. Numerical examples of approximating high-order derivatives using both the iterated and direct approach based on B-spline quasi-interpolation and multiquadric quasi-interpolation are presented at the end of the paper, which demonstrate that the iterated quasi-interpolation approach provides higher approximation orders than the corresponding direct approach. [ABSTRACT FROM AUTHOR]

In this paper, we propose a variational method in order to solve Bratu's problem for two dimensions in an adequate space of biquadratic spline functions. The solution is obtained by resolving a sequence of boundary value problems. We study some characterizations of the functions of such sequence and we express them as some linear combination of biquadratic spline bases functions. We finish by showing some numerical and graphical examples in order to prove the validity and the effectiveness of our method. [ABSTRACT FROM AUTHOR]

We discuss the problem of fitting a smooth regular curve γ : [ 0 , T ] → E n based on reduced data Q m = { q i } i = 0 m in arbitrary Euclidean space E n . The respective interpolation knots T = { t i } i = 0 m satisfying q i = γ (t i) are assumed to be unknown. In our setting the substitutes T λ = { t ^ i } i = 0 m of T are selected according to the so-called exponential parameterization governed by Q m and λ ∈ [ 0 , 1 ] . A modified Hermite interpolant γ ^ H introduced in Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004) is used here to fit (T ^ λ , Q m) . The case of λ = 1 (i.e. for cumulative chords) for general class of admissible samplings yields a sharp quartic convergence order in estimating γ ∈ C 4 by γ ^ H [see Kozera (Stud Inf 25(4B–61):1–140, 2004) and Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004)]. As recently shown in Kozera and Wilkołazka (Math Comput Sci, 2018. 10.1007/s11786-018-0362-4) the remaining λ ∈ [ 0 , 1) render a linear convergence order in γ ^ H ≈ γ for any Q m sampled more-or-less uniformly. The related analysis relies on comparing the difference γ - γ ^ H ∘ ϕ H in which ϕ H forms a special mapping between [0, T] and [ 0 , T ^ ] with T ^ = t ^ m . In this paper: (a) several sufficient conditions enforcing ϕ H to yield a genuine reparameterization are first formulated and then analytically and symbolically simplified. The latter covers also the asymptotic case expressed in a simple form. Ultimately, the reformulated conditions can be algebraically verified and/or geometrically visualized, (b) additionally in Sect. 3, the sharpness of the asymptotics of γ - γ ^ H ∘ ϕ H [from Kozera and Wilkołazka (Math Comput Sci, 2018. 10.1007/s11786-018-0362-4)] is proved upon applying symbolic and analytic calculations in Mathematica. [ABSTRACT FROM AUTHOR]

We present an approximation problem of surfaces of a generalized offset surface with offset variable distances and directions. Such approximating surface fits some given data points and minimizes a Sobolev's semi-norm of order 3. The study of the new results, from a mathematical point of view, carefully establishing the proof of the convergence between the generalized offset surface and its approximating spline in an adequate parametric bicubic spline space. Moreover, the approximating spline function is computed and an estimation of the relative error is introduced. Finally, some numerical and graphic examples are shown in order to prove the useful and the effectiveness of our method. [ABSTRACT FROM AUTHOR]

We study some procedures for the approximation of three-dimensional data on a grid with a hypothesis of periodicity. The first part proposes a generalization of a discrete periodic approximation defined by Dunham Jackson. The functions used have the advantage of owning an analytical explicit expression in terms of the samples (specific values) of the original function or data. In the second part, we describe a continuous approximation function for the same problem, defined through an integral. Some results of the rate of convergence and bounds of the approximation error are presented, with the single hypothesis of Hölder continuity or continuity of the original function. [ABSTRACT FROM AUTHOR]

This paper introduces the software TiGL: TiGL is an open source geometry modeler that is used in the conceptual and preliminary aircraft and helicopter design phase. It creates full three-dimensional models of aircraft from their parametric CPACS description. Due to its parametric nature, it is typically used for aircraft design analysis and optimization. First, we present the use-case and architecture of TiGL. Then, we discuss its geometry module, which is used to generate the B-spline based surfaces of the aircraft. The backbone of TiGL is its surface generator for curve network interpolation, based on Gordon surfaces. One major part of this paper explains the mathematical foundation of Gordon surfaces on B-splines and how we achieve the required curve network compatibility. Finally, TiGL's aircraft component module is introduced, which is used to create the external and internal parts of aircraft, such as wings, flaps, fuselages, engines or structural elements. [ABSTRACT FROM AUTHOR]

This work discusses the problem of fitting a regular curve γ based on reduced data points Q m = (q 0 , q 1 , ⋯ , q m) in arbitrary Euclidean space. The corresponding interpolation knots T = (t 0 , t 1 , ⋯ , t m) are assumed to be unknown. In this paper the missing knots are estimated by T λ = (t ^ 0 , t ^ 1 , ⋯ , t ^ m) in accordance with the so-called exponential parameterization (see Kvasov in Methods of shape-preserving spline approximation, World Scientific Publishing Company, Singapore, 2000) controlled by a single parameter λ ∈ [ 0 , 1 ] . In order to fit (T ^ λ , Q m) , a modified Hermite interpolant γ ^ H (a C 1 piecewise-cubic) introduced in Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004) is used. The sharp quartic convergence order for estimating γ ∈ C 4 by γ ^ H is proved in Kozera (Stud Inf 25(4B-61):1–140, 2004) and Kozera and Noakes (2004) only for λ = 1 and within the general class of admissible samplings. The main result of this paper extends the latter to the remaining cases of exponential parameterization covering all λ ∈ [ 0 , 1) . A slower linear convergence order in trajectory estimation is established for any λ ∈ [ 0 , 1) and arbitrary more-or-less uniform sampling. The numerical tests conducted in Mathematica indicate the sharpness of the latter and confirm the necessity of more-or-less uniformity. Other interpolation schemes used to fit reduced data Q m and based on T λ together with some relevant applications are also briefly recalled in this paper. [ABSTRACT FROM AUTHOR]

Numerical methods in approximation of derivatives on regular or irregular gridded and scattered data points are of significance in the study of numerous problems in engineering and geoscience. A mathematically accurate and stable method is assessed to compute the differential quantities, while being flexible and easy to implement. Strain-invariant parameters in deformation analysis and curvature attributes in geometrical analysis of approximated surfaces are independent quantities which could be derived numerically through radial basis functions (RBFs) for scattered data points. Approximation of a function or its derivatives through RBFs is highly correlated to its shape parameter which depends on the number and distribution of the particles on the support domain. A procedure to find the optimal shape parameter for RBFs in the influence domain of each nodal point together with the presentation of a manner of calculating the curvature attributes and strain parameters through Gaussian RBFs is proposed here. Results indicate a significant improvement in the approximation accuracy of strain parameters and curvature attributes, while this approximation is more stable when RBFs with optimal shape parameter are implemented rather than the traditional moving least squares. [ABSTRACT FROM AUTHOR]

The paper provides an approach for constructing multivariate radial kernels satisfying higher-order generalized Strang-Fix conditions from a given univariate generator. There are three key features of the approach. First, the kernels are explicitly expressed only by the derivatives of the f-form of the generator without computing any Fourier transforms. Second, it includes the radial kernels with the highest-order generalized Strang-Fix conditions. Finally, it requires only computing univariate derivatives of the f-form. Therefore, the approach is simple, efficient and easy to implement. As examples, the paper constructs radial kernels from some commonly used generators, including the Gaussian functions, the inverse multiquadric functions and compactly supported positive definite functions. [ABSTRACT FROM AUTHOR]

The objective of this article is to introduce a generalized algorithm to produce the m-point n-ary approximating subdivision schemes (for any integer m, n ≥ 2). The proposed algorithm has been derived from uniform B-spline blending functions. In particular, we study statistical and geometrical/traditional methods for the model selection and assessment for selecting a subdivision curve from the proposed family of schemes to model noisy and noisy free data. Moreover, we also discuss the deviation of subdivision curves generated by proposed family of schemes from convex polygonal curve. Furthermore, visual performances of the schemes have been presented to compare numerically the Gibbs oscillations with the existing family of schemes. [ABSTRACT FROM AUTHOR]

The piecewise interpolation method for constructing a smooth curve passing through the sequence of points in the plane is presented. For each section, an interpolation between two points is done by a rational function describing the curve of Agnesi. Conditions of existence of the curve on a section and the methodology for determining its parameters are established. The interpolation error is estimated. The proposed scheme for $$C^1$$ rational interpolation can be used to solve the problems of shape preserving interpolation. The described algorithm can be adapted for interpolation of convex, monotonous, or positive datasets. Examples of numeric data are given. [ABSTRACT FROM AUTHOR]

In this paper we present a Bayesian framework for interpolating data in a reproducing kernel Hilbert space associated with a random subdivision scheme, where not only approximations of the values of a function at some missing points can be obtained, but also uncertainty estimates for such predicted values. This random scheme generalizes the usual subdivision by taking into account, at each level, some uncertainty given in terms of suitably scaled noise sequences of i.i.d. Gaussian random variables with zero mean and given variance, and generating, in the limit, a Gaussian process whose correlation structure is characterized and used for computing realizations of the conditional posterior distribution. The hierarchical nature of the procedure may be exploited to reduce the computational cost compared to standard techniques in the case where many prediction points need to be considered. [ABSTRACT FROM AUTHOR]

This paper is concerned with the modeling of engineering images by the fractal properties of 6-point binary interpolating scheme. Association between the fractal behavior of the limit curve/surface and the parameter is obtained. The relationship between the subdivision parameter and the fractal dimension of the limit fractal curve of subdivision fractal is also presented. Numerical examples and visual demonstrations show that 6-point scheme is good choice for the generation of fractals for the modeling of fractal antennas, bearings, garari's and rock etc. [ABSTRACT FROM AUTHOR]

In this paper two non-stationary forms of Chaikin’s perturbation subdivision scheme, mentioned in Dyn et al. (2004), have been proposed with tension parameter ω . Comparison among the proposed subdivision schemes and the existing non-stationary subdivision scheme depicts that the trigonometric form is more efficient in the reproduction of circles and ellipses and the hyperbolic form is more suitable for the construction of many analytical curves. [ABSTRACT FROM AUTHOR]

We generate various new radial kernels by taking derivatives of known kernels with respect to scale. This is different from the well-known scale mixtures used before. In addition, we provide a simple recipe that explicitly constructs new kernels from the negative Laplacian of known kernels. Surprisingly, these two methods for generating new kernels can be proven to coincide for certain standard classes of radial kernels. The resulting radial kernels are positive definite, and a few illustrations are provided. [ABSTRACT FROM AUTHOR]

In this paper we present a variational approximation method for solving Troesch's problem. The existence and the uniqueness of this problem are shown. Moreover, we construct a sequence of approximate solutions of the problem from the number of knots of a partition of the domain. Such sequence converges to the exact solution of the problem. Finally, we analyze some graphical and numerical examples in order to show the efficiency of our method. [ABSTRACT FROM AUTHOR]

This article presents a new model for pitch-and-rhythm interrelationships and demonstrates the importance of these interrelationships as features for music analysis and classification. We show that the model is capable of identifying certain musical patterns and categorising pieces created by Mozart's dice game by identifying their quality. We discuss the cognitive foundation of this model and present future directions and applications. We employ the subdivision-based music analysis methods introduced earlier by the present authors to model certain pitch-and-rhythm interdependencies and symmetry patterns in selected pieces, shedding more light on the analogy proposed between subdivision schemes and musical structures. Our approach to pitch-and-rhythm interactions points out an interesting phenomenon, according to which the relations between neighbour pitches may be related to their timing through a relatively simple model. We propose hierarchical reductions and indicate that their levels are interrelated through simple subdivision schemes using new criteria, yielding features for classification. [ABSTRACT FROM PUBLISHER]

Though a Hermite subdivision scheme is non-stationary by nature, its non-stationarity can be of two types, making useful the distinction between Inherently Stationary and Inherently Non-Stationary Hermite subdivision schemes. This paper focuses on the class of inherently stationary, dual non-interpolatory Hermite subdivision schemes that can be obtained from known Hermite interpolatory ones, by applying a generalization of the de Rham corner cutting strategy. Exploiting specific tools for the analysis of inherently stationary Hermite subdivision schemes we show that, giving up the interpolation condition, the smoothness of the associated basic limit function can be increased by one, while its support width is only enlarged by one. To accomplish the analysis of de Rham-type Hermite subdivision schemes two new theoretical results are derived and the new notion of $$HC^{\ell }$$ -convergence is introduced. It allows the construction of Hermite-type subdivision schemes of order $$d+1$$ with the first element of the vector valued limit function having regularity $$\ell \ge d$$ . [ABSTRACT FROM AUTHOR]

This paper presents a new method for the analysis of convergence and smoothness of univariate nonuniform subdivision schemes. The analysis involves ideas from the theory of asymptotically equivalent subdivision schemes and nonuniform Laurent polynomial representation together with a new perturbation result. Application of the new method is presented for the analysis of interpolatory subdivision schemes based upon extended Chebyshev systems and for a class of smoothly varying schemes. [ABSTRACT FROM AUTHOR]

The objective of the present paper is the study of affine transformations of the plane, which provide self-affine curves as attractors. The properties of these curves depend decisively of the coefficients of the system of affnities involved. The corresponding functions are continuous on a compact interval. If the scale factors are properly chosen one can define Schauder bases ofC[a,b] composed by affine fractal functions close to polygonals. They can be chosen bounded. The basis constants and the biorthogonal sequence of coefficient functionals are studied. [ABSTRACT FROM AUTHOR]

Subdivision schemes are multi-resolution methods used in computer-aided geometric design to generate smooth curves or surfaces. We propose two new models for data analysis and compression based on subdivision schemes:(a) The ‘subdivision regression’ model, which can be viewed as a special multi-resolution decomposition.(b) The ‘tree regression’ model, which allows the identification of certain patterns within the data. The paper focuses on analysis and mentions compression as a byproduct. We suggest applying certain criteria on the output of these models as features for data analysis. Differently from existing multi-resolution analysis methods, these new models and criteria provide data features related to the schemes (the filters) themselves, based on a decomposition of the data into different resolution levels, and they also allow analysing data of non-smooth functions and working with varying-resolution subdivision rules. Finally, applications of these methods for music analysis and other potential usages are mentioned. [ABSTRACT FROM PUBLISHER]

Fractal interpolation is a modern technique in approximation theory to fit and analyze scientific data. We develop a new class of $$\mathcal C ^1$$- rational cubic fractal interpolation functions, where the associated iterated function system uses rational functions of the form $$\frac{p_i(x)}{q_i(x)},$$ where $$p_i(x)$$ and $$q_i(x)$$ are cubic polynomials involving two shape parameters. The rational cubic iterated function system scheme provides an additional freedom over the classical rational cubic interpolants due to the presence of the scaling factors and shape parameters. The classical rational cubic functions are obtained as a special case of the developed fractal interpolants. An upper bound of the uniform error of the rational cubic fractal interpolation function with an original function in $$\mathcal C ^2$$ is deduced for the convergence results. The rational fractal scheme is computationally economical, very much local, moderately local or global depending on the scaling factors and shape parameters. Appropriate restrictions on the scaling factors and shape parameters give sufficient conditions for a shape preserving rational cubic fractal interpolation function so that it is monotonic, positive, and convex if the data set is monotonic, positive, and convex, respectively. A visual illustration of the shape preserving fractal curves is provided to support our theoretical results. [ABSTRACT FROM AUTHOR]

We analyse the mask associated with the $2n$-point interpolatory Dubuc–Deslauriers subdivision scheme $S_{a^{[n]}}$. Sharp bounds are presented for the magnitude of the coefficients $a^{[n]}_{2i-1}$ of the mask. For scales $i \in [1,\sqrt{n}]$ it is shown that $|a^{[n]}_{2i-1}|$ is comparable to $i^{-1}$, and for larger power scales, exponentially decaying bounds are obtained. Using our bounds, we may precisely analyse the summability of the mask as a function of $n$ by identifying which coefficients of the mask contribute to the essential behaviour in $n$, recovering and refining the recent result of Deng–Hormann–Zhang that the operator norm of $S_{a^{[n]}}$ on $\ell ^\infty $ grows logarithmically in $n$. [ABSTRACT FROM PUBLISHER]

The class of symmetric positive definite matrices is an important class both in theory and application. Although this class is well studied, little is known about how to efficiently interpolate such data within the class. We extend the 4-point interpolatory subdivision scheme, as a method of interpolation, to data consisting of symmetric positive definite matrices. This extension is based on an explicit formula for calculating a binary 'geodetic average'. Our method generates a smooth curve of matrices, which retain many important properties of the interpolated matrices. Furthermore, the scheme is robust and easy to implement. [ABSTRACT FROM AUTHOR]

This paper describes a method to smooth an arc spline. Arc splines areG1continuous segments made of circular arcs and straight lines. We have proposed a smooth version of the arc spline by replacing its parts with C-, S-, and J-shaped spiral transitions, stitched withG2continuity, by using asinglesegment of Pythagorean hodograph quintic function. Use of asinglepolynomial function rather than two has the benefit that designers have fewer entities to deal with. Spiral transitions are important in manufacturing industries because of their use in the cutting paths for numerically controlled cutting machinery, highway or railway designing, non-holonomic robot path planning and spur gear designing. [ABSTRACT FROM AUTHOR]

The paper is devoted to the problem of smooth approximation of data. We are concerned with the exact interpolation of the data at nodes and, at the same time, with the smoothness of the interpolating curve and its derivatives. The same procedure is applied to the fitting of data (smoothing of data), too. The approximating curve is defined as the solution of a variational problem with constraints (interpolation conditions at nodes, data fitting property of the curve) the existence of whose unique solution we prove. Except for the constraints, the formulation of the problem of data approximation is not unique in general as our requirements on the behavior of the approximating curve between the nodes can be very subjective. The smooth approximation of this kind can lead, e.g., to the cubic spline interpolation.We discuss the choice of basis systems for this way of approximation and present results of several 1D numerical examples that show some advantages and drawbacks of smooth approximation. [ABSTRACT FROM AUTHOR]

Quantum splines are piecewise polynomials whose quantum derivatives (i.e. certain discrete derivatives or equivalently certain divided differences) agree up to some order at the joins. Just like classical splines, quantum splines admit a canonical basis with compact support: the quantum B-splines. These quantum B-splines are the q-analogues of classical B-splines. Here quantum B-spline bases and quantum B-spline curves are investigated, using a new variant of the blossom: the q ( quantum)- blossom. The q-blossom of a degree d polynomial is the unique symmetric, multiaffine function in d variables that reduces to the polynomial along the q-diagonal. By applying the q-blossom, algorithms and identities for quantum B-spline bases and quantum B-spline curves are developed, including quantum variants of the de Boor algorithms for recursive evaluation and quantum differentiation, knot insertion procedures for converting from quantum B-spline to piecewise quantum Bézier form, and a quantum variant of Marsden's identity. [ABSTRACT FROM AUTHOR]

We construct trivariate $$C^r$$ macro-elements for any $$r\ge 1$$ over the Worsey-Farin refinement of any tetrahedral partition. This extends the construction of $$C^1$$ cubic Worsey-Farin elements and $$C^2$$ elements of degree nine to the $$C^r$$ situation with $$r>2$$. We also show that the degree of polynomials used for our macro-elements is the lowest. [ABSTRACT FROM AUTHOR]

Subdivision schemes are multi-resolution methods used in computer-aided geometric design to generate smooth curves or surfaces. In this paper, we are interested in both smooth and non-smooth subdivision schemes. We propose two models that generalize the subdivision operation and can yield both smooth and non-smooth schemes in a controllable way: (1) The ‘varying-resolution’ model allows a structured access to the various resolutions of the refined data, yielding certain patterns. This model generalizes the standard subdivision iterative operation and has interesting interpretations in the geometrical space and also in creativity-oriented domains, such as music. As an infrastructure for this model, we propose representing a subdivision scheme by two dual rules trees. The dual tree is a permuted rules tree that gives a new operator-oriented view on the subdivision process, from which we derive an ‘adjoint scheme’. (2) The ‘generalized perturbed schemes’ model can be viewed as a special multi-resolution representation that allows a more flexible control on adding the details. For this model, we define the terms ‘template mask’ and ‘tension vector parameter’. The non-smooth schemes are created by the permutations of the ‘varying-resolution’ model or by certain choices of the ‘generalized perturbed schemes’ model. We then present procedures that integrate and demonstrate these models and some enhancements that bear a special meaning in creative contexts, such as music, imaging and texture. We describe two new applications for our models: (a) data and music analysis and synthesis, which also manifests the usefulness of the non-smooth schemes and the approximations proposed, and (b) the acceleration of convergence and smoothness analysis, using the ‘dual rules tree’. [ABSTRACT FROM AUTHOR]

This paper develops a smoothing domain-based energy (SDE) error indicator and an efficient adaptive procedure using edge-based smoothed point interpolation methods (ES-PIM), in which the strain field is constructed via the generalized smoothing operation over smoothing domains associated with edges of three-node triangular background cells. Because the ES-PIM can produce a close-to-exact stiffness and achieve 'super-convergence' and 'ultra-accurate' solutions, it is an ideal candidate for adaptive analysis. A SDE error indicator is first devised to make use of the features of the ES-PIM. A local refinement technique based on the Delaunay algorithm is then implemented to achieve high efficiency. The refinement of nodal neighbourhood is accomplished simply by adjusting a scaling factor assigned to control local nodal density. Intensive numerical studies, including the problems with stress concentration and solution singularity, demonstrate that the proposed adaptive procedure is effective and efficient in producing solutions with desired accuracy. [ABSTRACT FROM AUTHOR]

In this paper, two sextic-spline collocation methods are developed and analysed for approximating solutions of nonlinear fifth-order boundary-value problems. The first method uses a spline interpolant and the second one is based on a spline quasi-interpolant, which are constructed from sextic splines. They are both proved to be second-order convergent. Numerical results confirm the order of convergence predicted by the analysis. It has been observed that the methods developed in this paper are better than the others given in the literature. [ABSTRACT FROM AUTHOR]