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Normalized B-spline-like representation for low-degree Hermite osculatory interpolation problems

M. Boushabi, S. Eddargani, M. J. Ibáñez, A. Lamnii

TL;DR

This paper tackles Hermite osculatory interpolation on a refined univariate partition to enable low-degree splines while preserving high-order vertex smoothness. It develops a normalized B-spline-like basis for the MD-spline space $\\mathcal{S}^{1}(\\varphi, \\overline{\\tau}_n)$ using a geometric construction tied to Bernstein-B\\ezier blossoming and derives Marsden-type identities to build quasi-interpolants. Theoretical results guarantee nonnegativity, partition of unity, and locality of the basis, and the quasi-interpolants reproduce polynomials up to a degree $\\min_i \\varphi(i)$. Numerical experiments on several test functions demonstrate accuracy and robustness, with refinements enabling shape-preserving behavior and improved stability over existing approaches. Overall, the work advances univariate MD-spline representations and practical low-degree Hermite interpolation via normalized B-spline-like bases and Marsden-based quasi-interpolation.

Abstract

This paper deals with Hermite osculatory interpolating splines. For a partition of a real interval endowed with a refinement consisting in dividing each subinterval into two small subintervals, we consider a space of smooth splines with additional smoothness at the vertices of the initial partition, and of the lowest possible degree. A normalized B-spline-like representation for the considered spline space is provided. In addition, several quasi-interpolation operators based on blossoming and control polynomials have also been developed. Some numerical tests are presented and compared with some recent works to illustrate the performance of the proposed approach.

Normalized B-spline-like representation for low-degree Hermite osculatory interpolation problems

TL;DR

This paper tackles Hermite osculatory interpolation on a refined univariate partition to enable low-degree splines while preserving high-order vertex smoothness. It develops a normalized B-spline-like basis for the MD-spline space using a geometric construction tied to Bernstein-B\\ezier blossoming and derives Marsden-type identities to build quasi-interpolants. Theoretical results guarantee nonnegativity, partition of unity, and locality of the basis, and the quasi-interpolants reproduce polynomials up to a degree . Numerical experiments on several test functions demonstrate accuracy and robustness, with refinements enabling shape-preserving behavior and improved stability over existing approaches. Overall, the work advances univariate MD-spline representations and practical low-degree Hermite interpolation via normalized B-spline-like bases and Marsden-based quasi-interpolation.

Abstract

This paper deals with Hermite osculatory interpolating splines. For a partition of a real interval endowed with a refinement consisting in dividing each subinterval into two small subintervals, we consider a space of smooth splines with additional smoothness at the vertices of the initial partition, and of the lowest possible degree. A normalized B-spline-like representation for the considered spline space is provided. In addition, several quasi-interpolation operators based on blossoming and control polynomials have also been developed. Some numerical tests are presented and compared with some recent works to illustrate the performance of the proposed approach.
Paper Structure (12 sections, 12 theorems, 54 equations, 4 figures, 6 tables)

This paper contains 12 sections, 12 theorems, 54 equations, 4 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Normalized B-spline-like representation
  4. Classical Hermite basis
  5. Normalized B-spline-like basis
  6. B-spline-like representation
  7. Quasi-interpolants schemes
  8. Differential quasi-interpolation operator
  9. Quasi-interpolation based on point values
  10. Discrete quasi-interpolation operator based on polarization
  11. Numerical results
  12. Conclusion

Key Result

Lemma 1

The spline $s$ is $\mathcal{C}^{1}$ smooth at $\zeta_{i}$ if and only if where ${c}_{\alpha}$ and $\hat{c}_{\alpha}$ are the B-ordinates of $\mathit{p}_{1}$ and $\mathit{p}_{2}$, respectively.

Figures (4)

  • Figure 1: A schematic representation of the B-ordinates of $s$ relative to the subintervals $[v_{i},\zeta_{i}]$ and $[\zeta_{i},v_{i+1}]$.
  • Figure 2: Examples of B-spline-like functions associated with an interior vertex $v_i$: (left) The three basis functions corresponding to problem (\ref{['main_problem_interpolation']}) with $\varphi(i)=3$, $\varphi(i-1)=2$ and $\varphi(i+1)=4$. (right) The four basis functions corresponding to problem (\ref{['main_problem_interpolation']}) with $\varphi(i)=4, \varphi(i-1)=2, \varphi(i+1)=7$.
  • Figure 3: Plots of the boundary B-spline-like functions outlined in Figure \ref{['figure_interior']}: (Top, left) the case of three data at $v_0$ and four data at $v_1$. (Top, right) two data at $v_{n-1}$ and three at $v_n$. (Bottom, left) the case of four data at $v_0$ and seven at $v_1$. (Bottom, right) two data at $v_{n-1}$ and four at $v_n$.
  • Figure 4: Plots of the test functions, from left to right: $f_1$, $f_2$ and $f_3$.

Theorems & Definitions (15)

  • Lemma 1
  • Lemma 2
  • Proposition 3
  • Theorem 4
  • Remark 1
  • Lemma 5
  • Lemma 6
  • Remark 2
  • Theorem 7
  • Remark 3
  • ...and 5 more