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A proof that Wronskians of non-zero B-splines are positive is given, using only recursive formulas for B-splines and their derivatives. This could be used to generalize the de Boor–DeVore geometric proof of the Schoenberg–Whitney conditions and total positivity of B-splines to Hermite interpolation. For Wronskians of maximal order with respect to a given degree, positivity follows from a simple formula. [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]

In this study, the numerical solutions of the modified Fornberg–Whitham (mFW) equation, which describes immigration of the solitary wave and peakon waves with discontinuous first derivative at the peak, have been obtained by the collocation finite element method using quintic trigonometric B-spline bases. Although there are solutions of this equation by semi-analytical and analytical methods in the literature, there are very few studies on the solution of the equation by numerical methods. Any linearization technique has not been used while applying the method. The stability analysis of the applied method is examined by the von-Neumann Fourier series method. To show the performance of the method, we have considered three test problems with nonhomogeneous boundary conditions having analytical solutions. The error norms L2 and L∞ are calculated to demonstrate the accuracy and efficiency of the presented numerical scheme. [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]

In this paper, we address the problem of constructing C 2 cubic spline functions on a given arbitrary triangulation T . To this end, we endow every triangle of T with a Wang–Shi macro-structure. The C 2 cubic space on such a refined triangulation has a stable dimension and optimal approximation power. Moreover, any spline function in this space can be locally built on each of the macro-triangles independently via Hermite interpolation. We provide a simplex spline basis for the space of C 2 cubics defined on a single macro-triangle which behaves like a Bernstein/B-spline basis over the triangle. The basis functions inherit recurrence relations and differentiation formulas from the simplex spline construction, they form a nonnegative partition of unity, they admit simple conditions for C 2 joins across the edges of neighboring triangles, and they enjoy a Marsden-like identity. Also, there is a single control net to facilitate control and early visualization of a spline function over the macro-triangle. Thanks to these properties, the complex geometry of the Wang–Shi macro-structure is transparent to the user. Stable global bases for the full space of C 2 cubics on the Wang–Shi refined triangulation T are deduced from the local simplex spline basis by extending the concept of minimal determining sets. [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]

A cubic H-Bézier geometric approximation scheme with one free parameter τis introduced for parabolic arcs. It has four control points. The control points are computed by matching the end points and end unit tangents of the two curves and provide two more free parameters. These free parameters are computed by curvature continuity constraints. Optimal value of τ is obtained by minimizing the maximum value of the distance between the cubic H-Bézier curve and the concerned parabolic arc. The method is illustrated using different numerical examples which show that the proposed method is economical, reliable and efficient.

The existence of fundamental cardinal exponential B-splines of positive real order σ is established subject to two conditions on σ and their construction is implemented. A sampling result for these fundamental cardinal exponential B-splines is also presented. [ABSTRACT FROM AUTHOR]

We offer the algorithm for choosing tension parameters of the generalized splines for convexity preserving interpolation. The resulting spline minimally differs from the classical cubic spline and coincides with it if sufficient convexity conditions are satisfied for the last one. We consider specific algorithms for different generalized cubic splines such as rational, exponential, variable power, hyperbolic splines, and splines with additional knots. [ABSTRACT FROM AUTHOR]

This article deals with univariate binary approximating subdivision schemes and their generalization to non-tensor product bivariate subdivision schemes. The two algorithms are presented with one tension and two integer parameters which generate families of univariate and bivariate schemes. The tension parameter controls the shape of the limit curve and surface while integer parameters identify the members of the family. It is demonstrated that the proposed schemes preserve monotonicity of initial data. Moreover, continuity, polynomial reproduction and generation of the schemes are also discussed. Comparison with existing schemes is also given.

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]

In this paper, a proficient numerical technique for the time-fractional telegraph equation (TFTE) is proposed. The chief aim of this paper is to utilize a relatively new type of B-spline called the cubic trigonometric B-spline for the proposed scheme. This technique is based on finite difference formulation for the Caputo time-fractional derivative and cubic trigonometric B-splines based technique for the derivatives in space. A stability analysis of the scheme is presented to confirm that the errors do not amplify. A convergence analysis is also presented. Computational experiments are carried out in addition to verify the theoretical analysis. Numerical results are contrasted with a few present techniques and it is concluded that the presented scheme is progressively right and more compelling. [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]

This paper presents a computational technique based on cubic trigonometric B-splines for the time fractional Burgers' equation. The nonlinear advection term is approximated by a new linearization technique which is very efficient and significantly reduces the computational cost. The usual finite difference formulation is used to approximate the Caputo time fractional derivative while the derivative in space is discretized using cubic trigonometric B-spline functions. The proposed technique is proved to be globally unconditionally stable. A convergence analysis is discussed to measure the accuracy of the solution. Computational experiments are performed to further confirm the accuracy and stability of the method. Numerical results are compared with those obtained by a scheme based on parametric spline functions. The comparison reveals that the proposed scheme is quite accurate and effective. [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]

In this paper, we study how to construct a planar cubic G1 Hermite interpolation curve with minimal length and curvature variation energy. This problem is solved by a bi-objective optimization model. We prove that there is a unique solution to this problem and give the expression of the solution. Some numerical examples show that the proposed method makes the curvature variation energy of the curve as small as possible when the length of the curve as short as possible. [ABSTRACT FROM AUTHOR]

In the current paper, authors proposed a computational model based on the cubic B-spline method to solve linear 6th order BVPs arising in astrophysics. The prescribed method transforms the boundary problem to a system of linear equations. The algorithm we are going to develop in this paper is not only simply the approximation solution of the 6th order BVPs using cubic B-spline, but it also describes the estimated derivatives of 1st order to 6th order of the analytic solution at the same time. This novel technique has lesser computational cost than numerous other techniques and is second order convergent. To show the efficiency of the proposed method, four numerical examples have been tested. The results are described using error tables and graphs and are compared with the results existing in the literature. [ABSTRACT FROM AUTHOR]

From spline theory it is well-known that univariate cubic spline interpolation, if carried out in its natural Hilbert space W 2 2 [ a , b ] and on point sets with fill distance h , converges only like O ( h 2 ) in L 2 [ a , b ] if no additional assumptions are made. But superconvergence up to order h 4 occurs if more smoothness is assumed and if certain additional boundary conditions are satisfied. This phenomenon was generalized in 1999 to multivariate interpolation in Reproducing Kernel Hilbert Spaces on domains Ω ⊂ R d for continuous positive definite Fourier-transformable shift-invariant kernels on R d . But the sufficient condition for superconvergence given in 1999 still needs further analysis, because the interplay between smoothness and boundary conditions is not clear at all. Furthermore, if only additional smoothness is assumed, superconvergence is numerically observed in the interior of the domain, but a theoretical foundation still is a challenging open problem. This paper first generalizes the “improved error bounds” of 1999 by an abstract theory that includes the Aubin–Nitsche trick and the known superconvergence results for univariate polynomial splines. Then the paper analyzes what is behind the sufficient conditions for superconvergence. They split into conditions on smoothness and localization , and these are investigated independently. If sufficient smoothness is present, but no additional localization conditions are assumed, it is numerically observed that superconvergence always occurs in the interior of the domain, and some supporting arguments are provided. If smoothness and localization interact in the kernel-based case on R d , weak and strong boundary conditions in terms of pseudodifferential operators occur. A special section on Mercer expansions is added, because Mercer eigenfunctions always satisfy the sufficient conditions for superconvergence. Numerical examples illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this note, we develop a local construction of cubic splines and make a comparative analysis of local integro cubic splines. We also derive explicit formulae for a local integro cubic spline and its first two derivatives. These formulae are short and four-point ones that require less computational cost compared to an integro cubic spline quasi-interpolant. [ABSTRACT FROM AUTHOR]

Some families of Haar spaces in ℝd,d≥1, whose basis functions are d-variate piecewise polynomials, are highlighted. The starting point is a sequence of univariate piecewise polynomials, called Lobachevsky splines, arised in probability theory and asymptotically related to the normal density function. Then, it is shown that d-variate Lobachevsky splines can be expressed as products of Lobachevsky splines. All these splines have simple analytic expressions and subsets of them are suitable for scattered data interpolation, allowing efficient computation and plain error analysis. [ABSTRACT FROM AUTHOR]

We consider the wide class of all piecewise Chebyshevian splines with connection matrices at the knots. We prove that a spline space of this class is “good for interpolation” if and only if the spline space obtained by integration is “good for design”. As a consequence, this provides us with a simple practical description of all such spline spaces which can be used for solving Hermite interpolation problems. These results strongly rely on the properties of blossoms. [ABSTRACT FROM AUTHOR]

The spline theory can be derived from two sources: the algebraic one (where splines are understood as piecewise smooth functions satisfying some continuity conditions) and the variational one (where splines are obtained via minimization of some quadratic functionals with constraints). We show that the general variational approach called smooth interpolation introduced by Talmi and Gilat covers not only the cubic spline but also the tension spline (called also spline in tension or spline with tension) in one or more dimensions. We show the results of a 1D numerical example that present the advantages and drawbacks of the tension spline. [ABSTRACT FROM AUTHOR]

Bernstein-Bézier techniques for analyzing polynomial spline fields in n variables and their divergence are developed. Dimension and a minimal determining set for continuous piecewise divergence-free spline fields on the Alfeld split of a simplex in ℝ are obtained using the new techniques, as well as the dimension formula for continuous piecewise divergence-free splines on the Alfeld refinement of an arbitrary simplicial partition in ℝ. [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]

The purpose of image registration is to determine a transformation such that the transformed version of the source image is similar to the target one. In this paper, we focus on landmark-based image registration using radial basis functions transformations, in particular on the topology preservation of compactly supported radial basis functions transformations. In Cavoretto and De Rossi (Appl Math Inf Sci 7:2113-2121, 2013), the performances of Gneiting's and Wu's functions are compared with the ones of other well-known schemes in image registration, as thin-plate spline and Wendland's functions. Several numerical experiments and real-life cases with medical images show differences in accuracy and smoothness of the considered interpolation methods, which can be explained taking into account their topology preservation properties. Here we analyze analytically and experimentally the topology preservation performances of Gneiting's functions, comparing results with the ones obtained in Yang et al. (Pattern Recognit Lett 32:1162-1177, 2011), where Wendland's and Wu's functions are considered. [ABSTRACT FROM AUTHOR]

The convergence rate of a multigrid method depends on the properties of the smoother and the so-called grid transfer operator. In this paper we define and analyze new grid transfer operators with a generic cutting size which are applicable for high order problems. We enlarge the class of available geometric grid transfer operators by relating the symbol analysis of the coarse grid correction with the approximation properties of univariate subdivision schemes. We show that the polynomial generation property and stability of a subdivision scheme are crucial for convergence and optimality of the corresponding multigrid method. We construct a new class of grid transfer operators from univariate primal binary and ternary pseudo-spline symbols. Our numerical results illustrate the behavior of the new grid transfer operators and provide promising preliminary results for the bivariate case. [ABSTRACT FROM AUTHOR]

In this paper existence and uniqueness of twelfth degree spline is proved with application to quadrature. This formula is in the class of splines of degree 12 and continuity order $$C^{12}$$ that matches the derivatives up to order 6 at the knots of a uniform partition. Some mistakes in the literature are pointed out and corrected. Numerical examples are given to illustrate the applicability and efficiency of the new method. [ABSTRACT FROM AUTHOR]

A family of smooth rational spline fractal interpolation function (SRFIF) is presented with the help of classical rational cubic spline. Each SRFIF of the family is identified uniquely by the values of vertical scaling factorand shape parametersand, and the shape of the fractal interpolating curves can be constrained and modified by selecting suitable parameters and vertical scaling factor. In order to meet the needs of practical design, a new method is developed to visualize constrained data in the view of constrained curves. In the special case, the methods of linear constraint and quadratic constraint are discussed. Also, convergence analysis shows that SRFIF gives a good approximation to the original function. [ABSTRACT FROM AUTHOR]

The $$ G^{2} $$ and $$ G^{1} $$ -approximation schemes are introduced to approximate the popular generalized Cornu spirals with the help of the parametric rational cubic trigonometric Bézier curves. The $$ G^{2} $$ -approximation scheme has two free parameters whereas $$ G^{1} $$ -approximation scheme has four free parameters. To approximate the generalized Cornu spirals, the values of these free parameters are optimized by the minimization of the maximum relative curvature error of approximation. By comparing the relative curvature errors of approximation schemes, the developed approximation schemes are found less erroneous and more efficient than the existing GCS approximation schemes. [ABSTRACT FROM AUTHOR]

In this paper some new methods for curvature approximation of circular arcs by low-degree Bézier curves are presented. Interpolation by geometrically continuous ( G1) parametric polynomials is considered. All derived approximants are given explicitly and are therefore practically applicable. Moreover, obtained results indicate that G1 biarcs with at least G1 continuity at the junction have smaller curvature error as parametric polynomial counterparts of the same degree. It is also shown that all considered methods provide optimal asymptotic approximation order. [ABSTRACT FROM AUTHOR]

Given points P 1 , P 2 , … , P n in the plane, we are concerned with the problem of finding a fair curve which interpolates the points. We assume that we have a method in hand, called a basic curve method, for solving the geometric Hermite interpolation problem of fitting a regular C ∞ curve between two points with prescribed tangent directions at the endpoints. We also assume that we have an energy functional which defines the energy of any basic curve. Using this basic curve method repeatedly, we can then construct G 1 curves which interpolate the given points P 1 , P 2 , … , P n . The tangent directions at the interpolation points are variable and the idea is to choose them so that the energy of the resulting curve (i.e., the sum of the energies of its pieces) is minimal. We give sufficient conditions on the basic curve method, the energy functional, and the interpolation points for (a) existence, (b) G 2 regularity, and (c) uniqueness of minimal energy interpolating curves. We also identify a one-parameter family of basic curve methods, based on parametric cubics, whose minimal energy interpolating curves are unique and G 2 under suitable conditions. One member of this family looks very promising and we suggest its use in place of conventional C 2 parametric cubic splines. [ABSTRACT FROM AUTHOR]

A $$C^1$$ rational quadratic trigonometric interpolating function with three parameters is developed. The order of approximation of the developed $$C^1$$ trigonometric interpolant is $$O( {h_i^2 })$$ . The $$C^1$$ rational quadratic trigonometric function is extended to a $$C^1$$ bivariate rational quadratic trigonometric function. The developed $$C^1$$ bivariate trigonometric interpolant has six parameters in each rectangular patch. Automatic selection schemes for parameters are developed to preserve the positive shape of curve and surface data using $$C^1$$ rational quadratic trigonometric function and $$C^1$$ bivariate rational quadratic trigonometric function respectively. Performing the result, it is noticed that developed positivity preserving interpolation schemes are fast, efficient and well suited for all data types. Several numerical examples are presented to ascertain the correctness and usability of developed schemes. [ABSTRACT FROM AUTHOR]

Constructing shape-preserving interpolation spline has been a hot topic in industrial design and scientific data visualization during the past 30 years. In the existing shape preservinginterpolation spline methods, however, some methods can be only used to preserve the monotonic data set, while others can be only used to preserve the convex data set, and often forcontinuity, it is requested to solve a linear system of consistency equations for the derivative values at the knots. In this paper, a new explicit representation of arational quartic interpolation spline with two local tension parameters is developed. A convergence analysis establishes an error bound and shows that the order of approximation isaccuracy. Sufficient conditions for the proposed interpolation spline to preserve the shape of positive, monotonic, and convex set of data are derived. [ABSTRACT FROM AUTHOR]

A $$GC^{1}$$ bi-quadratic trigonometric interpolation scheme with four free parameters in each rectangular patch is developed. The free parameters are constrained to avoid unnecessary oscillations, resulting in smooth positive and monotone surfaces for the positive and monotone 3D data. The formulated schemes are local and the degree of the interpolant is unique over the whole domain. [ABSTRACT FROM AUTHOR]

In this paper, we construct a numerical method based on cubic splines in tension for solving regularized long wave equation. The truncation error is analysed and the method shows that by choosing suitably parameters we can obtain various accuracy schemes. Numerical stability of the method has been studied by using a linearized stability analysis. Test problems are dealt with. The numerical simulations can validate and demonstrate the advantages of the method. [ABSTRACT FROM AUTHOR]

Martensen splines M f of degree n interpolate f and its derivatives up to the order n − 1 at a subset of the knots of the spline space, have local support and exactly reproduce both polynomials and splines of degree ≤ n . An approximation error estimate has been provided for f ∈ C n + 1 . This paper aims to clarify how well the Martensen splines M f approximate smooth functions on compact intervals. Assuming that f ∈ C n − 1 , approximation error estimates are provided for D j f , j = 0 , 1 , … , n − 1 , where D j is the j th derivative operator. Moreover, a set of sufficient conditions on the sequence of meshes are derived for the uniform convergence of D j M f to D j f , for j = 0 , 1 , … , n − 1 . [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]