Bivariate Lagrange interpolation at the Padua points: the ideal theory approach

Motivated by polynomial approximations of differential forms, we study analytical and numerical properties of a polynomial interpolation problem that relies on function averages over interval segments. The usage of segment data gives rise to new theoretical and practical aspects that distinguish this problem considerably from classical nodal interpolation. We will analyse fundamental mathematical properties of this problem as existence, uniqueness and numerical conditioning o...

Interpolation by various types of splines is the standard procedure in many applications. In this paper we shall discuss harmonic spline "interpolation" (on the lines of a grid) as an alternative to polynomial spline interpolation (at vertices of a grid). We will discuss some advantages and drawbacks of this approach and present the asymptotics of the $L_p$-error for adaptive approximation by harmonic splines.

An interpolation problem related to the elliptic Painlev\'e equation is formulated and solved. A simple form of the elliptic Painlev\'e equation and the Lax pair are obtained. Explicit determinant formulae of special solutions are also given.

The polynomial kernels are widely used in machine learning and they are one of the default choices to develop kernel-based classification and regression models. However, they are rarely used and considered in numerical analysis due to their lack of strict positive definiteness. In particular they do not enjoy the usual property of unisolvency for arbitrary point sets, which is one of the key properties used to build kernel-based interpolation methods. This paper is devoted to...

First, an abstract scheme of constructing biorthogonal rational systems related to some interpolation problems is proposed. We also present a modification of the famous step-by-step process of solving the Nevanlinna-Pick problems for Nevanlinna functions. The process in question gives rise to three-term recurrence relations with coefficients depending on the spectral parameter. These relations can be rewritten in the matrix form by means of two Jacobi matrices. As a result, a...

We describe some new univariate spline quasi-interpolants on uniform partitions of bounded intervals. Then we give some applications to numerical analysis: integration, differentiation and approximation of zeros.

In contrast to the univariate case, interpolation with polynomials of a given maximal total degree is not always possible even if the number of interpolation points and the space dimension coincide. Due to that, numerous constructions for interpolation sets have been devised, the most popular ones being based on intersections of lines. In this paper, we study algebraic properties of some such interpolation configurations, namely the approaches by Radon-Berzolari and Chung-Yao...

We describe an algorithm for controlling the relative error in the numerical evaluation of a bivariate integral, without prior knowledge of the magnitude of the integral. In the event that the magnitude of the integral is less than unity, absolute error control is preferred. The underlying quadrature rule is positive-weight interpolatory and composite. Some numerical examples demonstrate the algorithm.

The $\ell_2-$ and $\ell_1-$regularized modified Lagrange interpolation formulae over $[-1,1]$ are deduced in this paper. This paper mainly analyzes the numerical characteristics of regularized barycentric interpolation formulae, which are presented in [C. An and H.-N. Wu, 2019], and regularized modified Lagrange interpolation formulae for both interpolation and extrapolation. Regularized barycentric interpolation formulae can be carried out in $\mathcal{O}(N)$ operations base...

We present a new closed form for the interpolating polynomial of the general univariate Hermite interpolation that requires only calculation of polynomial derivatives, instead of derivatives of rational functions. This result is used to obtain a new simultaneous polynomial division by a common divisor over a perfect field. The above findings are utilized to obtain a closed formula for the semi--simple part of the Jordan decomposition of a matrix. Finally, a number of new iden...