Fundamental Splines Research Articles

Equivalence of the Fundamental Spline and Green’s Function in Constructing an Exact Finite-Dimensional Analog of the Boundary-Value Problem for an Ordinary Differential Equation of the Fourth Order

Equivalence of the Fundamental Spline and Green’s Function in Constructing an Exact Finite-Dimensional Analog of the Boundary-Value Problem for an Ordinary Differential Equation of the Fourth Order

Fundamental functions for local interpolation of quadrilateral meshes with extraordinary vertices

The aim of this work is to present a general and simple strategy for the construction of compactly supported fundamental spline (piecewise-polynomial) functions for local interpolation, that are defined over quadrangulations of the real plane with extraordinary vertices. The proposed strategy — which extends the univariate framework introduced in Antonelli et al. (Adv Comput Math 40:945–976, 2014) and Beccari et al. (J Comput Appl Math 240:5–19, 2013)—consists in considering a suitable combination of bivariate polynomial interpolants with blending functions that are the natural generalization of odd-degree tensor-product B-splines. These blending functions are constructed as basic limit functions of the bivariate, primal subdivision schemes developed simultaneously in Stam (Comput Aided Geom Des 18:397–427, 2001) and Zorin et al. (Comput Aided Geom Des 18:483–502, 2001). As an application example of our constructive strategy we present the compactly supported C^2 fundamental functions for local interpolation that arise by considering as blending functions the basic limit functions of the celebrated Catmull–Clark subdivision scheme proposed in Catmull and Clark (Comput Aided Des 46:103–124, 2016).

Properties of the fundamental splines of the high order

In this paper, the properties of the fundamental splines of high orders are numerically studied. From the point of view of the interpolation problem, the fundamental spline is a nodal function generated by a family of integer translations of the corresponding basic spline. We have established that with increasing order fundamental splines tend to the sampling function . Analogous assertions were obtained earlier for nodal functions on the basis of other systems of shifts. The behavior of the coefficients of basic splines is studied. With the help of calculations, it is shown that for n>2 there is a sign-reversal and a monotone decreasing modulo.

Quaternionic Fundamental Cardinal Splines: Interpolation and Sampling

B-splines $$B_{q}$$ of quaternionic order q, for short quaternionic B-splines, are quaternion-valued piecewise Muntz polynomials whose scalar parts interpolate the classical Schoenberg splines $$B_{n}$$ , $$n\in \mathbb {N}$$ , with respect to degree and smoothness. They in general do not satisfy the interpolation property $$B_{q}(n) = \delta _{n,0}$$ , $$n\in \mathbb {Z}$$ . However, the application of the interpolation filter $$(\sum \nolimits _{k\in \mathbb {Z}} \widehat{B_q}(\xi +2 \pi k))^{-1}$$ —if well-defined—in the frequency domain yields a cardinal fundamental spline of quaternionic order that satisfies the interpolation property. We handle the ambiguity of the quaternion-valued exponential function appearing in the denominator of the interpolation filter and relate the filter to interesting properties of a quaternionic Hurwitz zeta function and the existence of complex quaternionic inverses. Finally, we show that the cardinal fundamental splines of quaternionic order fit into the setting of Kramer’s Lemma and allow for a family of sampling, respectively, interpolation series.

Local Average Sampling and Reconstruction with Fundamental Splines of Fractional Order

We analyze sampling and average sampling techniques for fractional spline subspaces of Fractional B-splines are extensions of Schoenberg’s polynomial splines of integral order to real order We present the interpolation with fundamental splines of fractional order for and the average sampling with fundamental splines of fractional order for Further, we generalize Kramer’s lemma in the context of local average sampling.

ФОРМУВАННЯ ОРТОГОНАЛЬНИХ СІТОК НА ОСНОВІ ФУНДАМЕНТАЛЬНОГО СПЛАЙНУ

To model smooth curved contours the questions are raised about finding derivatives that can be formed based on user additional information. If such information does not exist, then additional numerical methods are used for calculation. To avoid extra calculations, it is relevant to use fundamental splines which are based only on the point frame information.To work with zero-length curves that allows the formation of orthogonal grids on a plane, additional research is needed in order to save this condition for different values ​​of parameter replacement. The author proposes a method of constructing flat orthogonal grids based on a flat isotropic fundamental spline. The conditions for creating a three-dimensional fundamental spline are given. To form a flat isotropic spline, the condition of isotropic chords is used, which subtend the point frame and forms the statement. It is proposed to use a quasi-conformal parameter change to form an orthogonal grid. The parameter is replaced by a combination of the defined functions f1(u), f2(v) - where u and v are some valid parameters. The functions must be formed on the basis of different parameters. The statement is given regarding the formation of an orthogonal grid based on an isotropic spline with the proposed parameter replacement. To prove the validity of the proposed states, examples of simulated grids are given and the coefficient of the first quadratic form F is calculated, the analysis of which allows us to make certain assumptions. Further studies are related to the construction of minimal surfaces based on the proposed quasi-conformal replacement. Keywords: isotropic curve, fundamental spline, orthogonal grid, isothermal grid, quasiconformal parameter replacement.

Mathematical models on the basis of fundamental trigonometric splines

The paper considers some systems of fundamental functions which are easy to be used in the problems for ordinary differential equations. The most well-known systems of fundamental functions include Lagrange system of fundamental functions and the complete systems even and odd fundamental trigonometric polynomials. However, there are some other systems of fundamental functions on analytical grid, including the systems of polynomial fundamental splines and complete trigonometric even and odd fundamental splines. Some of these systems are considered in this paper.

A general framework for the construction of piecewise-polynomial local interpolants of minimum degree

In this paper we consider the problem of designing piecewise polynomial local interpolants of non-uniformly spaced data. We provide a constructive approach that, for any assigned degree of polynomial reproduction, continuity order, and support width, allows for generating the fundamental spline functions of minimum degree having the desired properties. Finally, the proposed construction is extended to handle open sets of data and to the case of multiple knots.

Multivariate Interpolation with Fundamental Splines of Fractional Order

AbstractFractional B‐splines Bσ, σ ≥ 1, are piecewise polynomials of fractional degree that interpolate the classical Schoenberg splines $B_n, n \in {\rm I\!N}$, with respect to the degree. As the Schoenberg splines of order ≥ 3, they in general do not satisfy the interpolation property $B_\sigma(n-k) = \delta_{n,k},n,k \in {\rm Z\!\!Z}$. However, the application of the interpolation filter $1/\sum_{k\in{\rm Z\!\!Z}} \hat{B}_{\sigma}(\omega-2 \pi k)$—if well‐defined—in the frequency domain yields a fundamental spline of fractional order that does satisfy the interpolation property. We extend these result via ridge functions to multivariate fractional B‐splines. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Non-uniform interpolatory curve subdivision with edge parameters built upon compactly supported fundamental splines

In this paper we present a family of Non-Uniform Local Interpolatory (NULI) subdivision schemes, derived from compactly supported interpolatory fundamental splines with non-uniform knots (NULIFS). For this spline family, the knot-partition is defined by a sequence of break points and by one additional knot, arbitrarily placed along each knot-interval. The resulting refinement algorithms are linear and turn out to contain a set of edge parameters that, when fixed to a value in the range [0,1], allow us to achieve special shape features by simply moving each auxiliary knot between the break points. Among all the members of this new family of schemes, we will then especially analyze the NULI 4-point refinement. This subdivision scheme has all the fundamental features of the quadratic fundamental spline basis it is originated from, namely compact support, C1 smoothness, second order polynomials reproduction and approximation order 3. In addition the NULI 4-point subdivision algorithm has the possibility of setting consecutive edge parameters to simulate double and triple knots—that are not considered by the authors of the corresponding spline basis—thus allowing for limit curves with crease vertices, without using an ad hoc mask. Numerical examples and comparisons with other methods will be given to the aim of illustrating the performance of the NULI 4-point scheme in the case of highly non-uniform initial data.

Cardinal splines in nonparametric regression

We discuss a nonparametric regression model on an equidistant grid of the real line. A class of kernel type estimates based on the so-called fundamental cardinal splines will be introduced. Asymptotic optimality of these estimates will be established for certain functional classes. This model explains the often mentioned heuristic fact that cubic splines are adequate for most practical applications.

Splines in Higher Order TV Regularization

Splines play an important role as solutions of various interpolation and approximation problems that minimize special functionals in some smoothness spaces. In this paper, we show in a strictly discrete setting that splines of degree m?1 solve also a minimization problem with quadratic data term and m-th order total variation (TV) regularization term. In contrast to problems with quadratic regularization terms involving m-th order derivatives, the spline knots are not known in advance but depend on the input data and the regularization parameter ?. More precisely, the spline knots are determined by the contact points of the m---th discrete antiderivative of the solution with the tube of width 2? around the m-th discrete antiderivative of the input data. We point out that the dual formulation of our minimization problem can be considered as support vector regression problem in the discrete counterpart of the Sobolev space W 2,0 m . From this point of view, the solution of our minimization problem has a sparse representation in terms of discrete fundamental splines.

Lagrange Interpolation by Bivariate C1-Splines with Optimal Approximation Order

We develop the first local Lagrange interpolation scheme for C1-splines of degree q≥3 on arbitrary triangulations. For doing this, we use a fast coloring algorithm to subdivide about half of the triangles by a Clough–Tocher split in an appropriate way. Based on this coloring, we choose interpolation points such that the corresponding fundamental splines have local support. The interpolating splines yield optimal approximation order and can be computed with linear complexity. Numerical examples with a large number of interpolation points show that our method works efficiently.

The C r -fundamental splines of Clough–Tocher and Powell–Sabin types for Lagrange interpolation on a three direction mesh

Let Δ(1) be the uniform three direction mesh of the plane whose vertices are integer points of $$\mathbb{Z}^2 $$ .Let $$\Delta _C^{(1)} $$ (respectively $$\Delta _P^{(1)} $$ $$C^r (\mathbb{R}^2 )$$ of degree d=3r (respectively d=3r+1 ) for r odd (respectively even) on the triangulation $$\Delta _C^{(1)} $$ , and of degree d=2r (respectively d=2r+1) for r odd (respectively even) on the triangulation $$\Delta _P^{(1)} $$ . Using linear combinations of translates of these splines we obtain Lagrange interpolants whose corresponding order of approximation is optimal.

Applications of optimally local interpolation to interpolatory approximants and compactly supported wavelets

The objective of this paper is to introduce a general scheme for the construction of interpolatory approximation formulas and compactly supported wavelets by using spline functions with arbitrary (nonuniform) knots. Both construction procedures are based on certain “optimally local” interpolatory fundamental spline functions which are not required to possess any approximation property.

Interpolating and smoothing biquadratic spline

The paper deals with the biquadratic splines and their use for the interpolation in two variables on the rectangular mesh. The possibilities are shown how to interpolate function values, values of the partial derivative or values of the mixed derivative. Further, the so-called smoothing biquadratic splines are defined and the algorithms for their computation are described. All of these biquadratic splines are derived by means of the tensor product of the linear spaces of the quadratic splines and their bases are given by the so-called fundamental splines.

Cardinal Hermite Spline Interpolation with Shifted Nodes

Generalized cardinal Hermite spline interpolation is considered. A special case of this problem is the classical cardinal Hermite spline interpolation with shifted nodes. By means of a corresponding symbol new representations of the cardinal Hermite fundamental splines can be given. Furthermore, a new efficient algorithm for the computation of the cardinal Hermite spline interpolant is obtained, which is mainly based on fast Fourier transform. This algorithm is shown to be also applicable to computing the periodic Hermite spline interpolant. In both cases we only use necessary and sufficient conditions for the existence and uniqueness of the corresponding Hermite spline interpolant.

On Lp-Boundedness of the L2-Projector onto Splines

In this paper we offer a new approach to C. de Boor′s conjecture of the L∞-boundedness of the L2-projector PS onto the spline space Sm−1(Δn). This approach is based on the strengthening of the "exponential decay" property of the fundamental spline. It is proved, first, that the Lp-norm of the operator PS is uniformly bounded without any restrictions on the mesh Δn at least in some neighbourhood of p = 2 and, second, that the Lp-norm of the operator PS for all p ∈ [1, ∞] is uniformly bounded in meshes Δn with a fixed number of nodes n.

Cardinal Hermite-spline-interpolation on the equidistant lattice

The theory of Hermite-spline interpolation on the equidistant latticeZ is written in purly real terms and this for an arbitrary polynomial degree, Hermitian order and node-shift parameter. An explicit representation formula for the Hermitian fundamental splines (Lagrangians) is presented and the convergence of the corresponding Lagrange-series is discussed.

Zur konstruktion periodischer hermite-interpolationssplines bei äquidistanter knotenverteilung

A complex line integral representation of the fundamental splines is used as a starting point for a simple and lucid method for the construction of periodic interpolating Hermite-splines with equidistant knots. Several applications and generalizations are indicated.