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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 study, we investigate a hybrid-type anisotropic weakly over-penalised symmetric interior penalty method for the Poisson equation on convex domains. Compared with the well-known hybrid discontinuous Galerkin methods, our approach is simple and easy to implement. Our primary contributions are the proposal of a new scheme and the demonstration of a proof for the consistency term, which allows us to estimate the anisotropic consistency error. The key idea of the proof is to apply the relation between the Raviart–Thomas finite element space and a discontinuous space. In numerical experiments, we compare the calculation results for standard and anisotropic mesh partitions. [ABSTRACT FROM AUTHOR]

Based on Cauchy’s integral formula and conformal maps, this paper presents a new method for constructing barycentric rational interpolation formulae for complex functions, which may contain singularities such as poles, branch cuts, or essential singularities. The resulting interpolations are pole-free, exponentially convergent, and numerically stable, requiring only O (N) operations. Inspired by the logarithm equilibrium potential, we introduce a Möbius transform to concentrate nodes to the vicinity of singularity to get a spectacular improvement on approximation quality. A thorough convergence analysis is provided, alongside numerous numerical examples that illustrate the theoretical results and demonstrate the accuracy and efficiency of the methodology. Meanwhile, the paper also discusses some applications of the method including the numerical solutions of boundary value problems and the zero locations of holomorphic functions. [ABSTRACT FROM AUTHOR]

In this paper, a new Hermite interpolatory subdivision scheme for curve interpolation is introduced. The scheme is constructed from the Rational [3/2] Bernstein Bezier polynomial. We call it the [3/2]-scheme. The limit function of the [3/2]-scheme interpolates both the function values and their derivatives. The proposed scheme has three shape parameters w 0 , w 1 and w 2 . It is shown that if w 1 = w 0 + w 2 2 , then the [3/2]-scheme reproduces linear polynomial and is C 1 provided w 0 and w 2 lie in a region of convergence. The scheme also satisfies the shape preserving properties, i.e., monotonicity and convexity. We also compare the [3/2]-scheme with other existing schemes like the [2/2]-scheme and the Merrien scheme introduced recently. An error analysis shows that the [3/2]-scheme is better than the [2/2]-scheme and the Merrien scheme. Further, it is observed that in case w 0 = w 1 = w 2 , the [3/2]-scheme reduces to the Merrien scheme. [ABSTRACT FROM AUTHOR]

We present precise Raviart–Thomas interpolation error estimates on anisotropic meshes. The novel aspect of our theory is the introduction of a new geometric parameter of simplices. It is possible to obtain new anisotropic Raviart–Thoma error estimates using the parameter. We also include corrections to an error in “General theory of interpolation error estimates on anisotropic meshes” (Japan Journal of Industrial and Applied Mathematics, 38 (2021) 163-191), in which Theorem 3 was incorrect. [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]

We propose to extend results on the interpolation theory for scalar functions to the case of differential k-forms. More precisely, we consider the interpolation of fields in P r - Λ k (T) , the finite element spaces of trimmed polynomial k-forms of arbitrary degree r ≥ 1 , from their weights, namely their integrals on k-chains. These integrals have a clear physical interpretation, such as circulations along curves, fluxes across surfaces, densities in volumes, depending on the value of k. In this work, for k = 1 , we rely on the flexibility of the weights with respect to their geometrical support, to study different sets of 1-chains in T for a high order interpolation of differential 1-forms, constructed starting from "good" sets of nodes for a high order multi-variate polynomial representation of scalar fields, namely 0-forms. We analyse the growth of the generalized Lebesgue constant with the degree r and preliminary numerical results for edge elements support the nonuniform choice, in agreement with the well-known nodal case. [ABSTRACT FROM AUTHOR]

Classical rational interpolation usually enjoys better approximation properties than polynomial interpolation because it avoids wild oscillations and exhibits exponential convergence for approximating analytic functions. We develop a rational interpolation operator, which not only preserves the advantage of classical rational interpolation, but also has a finite Lebesgue constant. In particular, it is convergent for approximating any continuous function, and the convergence rate of the interpolants approximating a function is obtained using the modulus of continuity. [ABSTRACT FROM AUTHOR]

Cycloidal spaces are generated by the trigonometric polynomials of degree one and algebraic polynomials. The critical length of a cycloidal space is the supremum of the lengths of the intervals on which the Hermite interpolation problems are unisolvent. The critical length is related with the critical length for design purposes in computer-aided geometric design. This paper shows an unexpected connection of critical lengths with the zeros of Bessel functions. We prove that the half of the critical length of a cycloidal space is the first positive zero of a Bessel function of the first kind. [ABSTRACT FROM AUTHOR]

We construct a class of cubature formulae for harmonic functions on the unit disk based on line integrals over $$2n+1$$ distinct chords. These chords are assumed to have constant distance $$t$$ to the center of the disk, and their angles to be equispaced over the interval $$[0,2\pi ]$$ . If $$t$$ is chosen properly, these formulae integrate exactly all harmonic polynomials of degree up to $$4n+1$$ , which is the highest achievable degree of precision for this class of cubature formulae. For more generally distributed chords, we introduce a class of interpolatory cubature formulae which we show to coincide with the previous formulae for the equispaced case. We give an error estimate for a particular cubature rule from this class and provide numerical examples. [ABSTRACT FROM AUTHOR]

Fractal interpolation provides an efficient way to describe data that have smooth and non-smooth structures. Based on the theory of fractal interpolation functions (FIFs), the Hermite rational cubic spline FIFs (fractal boundary curves) are constructed to approximate an original function along the grid lines of interpolation domain. Then the blending Hermite rational cubic spline fractal interpolation surface (FIS) is generated by using the blending functions with these fractal boundary curves. The convergence of the Hermite rational cubic spline FIS towards an original function is studied. The scaling factors and shape parameters involved in fractal boundary curves are constrained suitably such that these fractal boundary curves are positive whenever the given interpolation data along the grid lines are positive. Our Hermite blending rational cubic spline FIS is positive whenever the corresponding fractal boundary curves are positive. Various collections of fractal boundary curves can be adapted with suitable modifications in the associated scaling parameters or/and shape parameters, and consequently our construction allows interactive alteration in the shape of rational FIS. [ABSTRACT FROM AUTHOR]

In this paper, we study two new spline methods for the Birkhoff interpolation problem proposed in Costabile and Longo (Calcolo 47:49-63, ). Our methods are based on cubic spline and quintic spline respectively. The new methods are easy to be implemented. They are effective not only in approximating the original function but also in approximating its derivatives. The respective remainders and error bounds are given. Numerical tests are performed. Numerical results confirm the theoretical analysis and show that our methods are very practical. [ABSTRACT FROM AUTHOR]

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 introduce a type of modified truncations of approximate approximation with Gaussian kernels, which can be applied to approximate functions on compact intervals. Our results improve the related results of Chen and Cao (Appl Math Comput 217:725-734, ), Müller and Varnhorn (J Approx Theory 145:171-181, ). Also, we construct approximate approximation operators in multivariate cases and obtain the estimates of approximation rate. [ABSTRACT FROM AUTHOR]

Discontinuous weighted least-squares (DWLS) approximation is modification of a weighted least-squares method that requires a local support (a reconstruction stencil) to approximate a function at a given point. A DWLS method is often employed in computational problems where a function is approximated on an irregular computational grid. It has recently been revealed that the method provides inaccurate approximation on irregular grids and conventional weighting of distant points captured by a reconstruction stencil on an irregular coarse mesh does not improve the accuracy of the approximation. Thus in our paper we further investigate the impact of distant points on the accuracy of DWLS approximation and design new weight coefficients for DWLS reconstruction that allow one to obtain more accurate reconstruction results. Our approach is based on a concept of numerically distant points originally developed in author's previous works, as a new weight function calculates the distance between two points in the data space. [ABSTRACT FROM AUTHOR]

A Birkhoff interpolation problem is studied. The explicit solution and estimation of remainder is given. The cases of equidistant and Chebyshev nodes are considered. This problem is of interest in collocation methods of boundary value problem (Costabile and Napoli in Far East J. Appl. Math. 25(3):289–305, ; Costabile and Longo in Applied and Industrial Mathematics in Italy III, , in press). In addition an application to quadrature is given. [ABSTRACT FROM AUTHOR]

We introduce an extension to the two-dimensional simplex of the univariate two-point expansion formula for sufficiently smooth real functions introduced in [13]; it is a polynomial expansion with algebraic degree of exactness. This expansion is applied to obtain a new class of embedded boundary-type cubature formulae on the simplex. [ABSTRACT FROM AUTHOR]

A method is considered for constructing weighted arithmetic means from IR to IR, which possess the interpolation property. Necessary and sufficient conditions are then given in order to solve the convergence problem to a continuous real function f(ξ), ξ∈IR of such interpolating mean functions. Finally a few formulas are considered as an application of the proposed method. [ABSTRACT FROM AUTHOR]

A class of inverse distance weighted formulas for scattered data interpolation, such as the well-known Shepard one, is reconsidered from a physical viewpoint and in particular with respect to electrostatic or gravitational fields. In this way, simple properties are explained, that permit the construction of either parallel or recursive algorithms to calculate these formulas. The formulas are then interpreted as means in accordance with the definitions of Cauchy, Chisini and others. As a consequence, some significant properties of the means can be used. Finally, a few comments are made on the numerical evaluation of the formulas considered. [ABSTRACT FROM AUTHOR]

A method to construct a wide class of weighted arithmetic means, possessing the interpolation property, is proposed. The approximation problem of a continuous real function by these interpolating mean functions is examined. A few examples are given and include the Shepard formula and the classical Hermite-Fejér formula. [ABSTRACT FROM AUTHOR]