Lebesgue Constants Research Articles

Pseudospectra of Matrix Polynomials that Are Expressed in Alternative Bases

Spectra and pseudospectra of matrix polynomials are of interest in geometric intersection problems, vibration problems, and analysis of dynamical systems. In this note we consider the effect of the choice of polynomial basis on the pseudospectrum and on the conditioning of the spectrum of regular matrix polynomials. In particular, we consider the direct use of the Lagrange basis on distinct interpolation nodes, and give a geometric characterization of “good” nodes. We also give some tools for computation of roots at infinity via a new, natural, reversal. The principal achievement of the paper is to connect pseudospectra to the well-established theory of Lebesgue functions and Lebesgue constants, by separating the influence of the scalar basis from the natural scale of the matrix polynomial, which allows many results from interpolation theory to be applied.

Bivariate Lagrange interpolation at the Padua points: Computational aspects

The so-called “Padua points” give a simple, geometric and explicit construction of bivariate polynomial interpolation in the square. Moreover, the associated Lebesgue constant has minimal order of growth O ( log 2 ( n ) ) . Here we show four families of Padua points for interpolation at any even or odd degree n , and we present a stable and efficient implementation of the corresponding Lagrange interpolation formula, based on the representation in a suitable orthogonal basis. We also discuss extension of (non-polynomial) Padua-like interpolation to other domains, such as triangles and ellipses; we give complexity and error estimates, and several numerical tests.

On the approximation of continuous functions of two variables

It is shown that if P m α,β (x) (α, β > −1, m = 0, 1, 2, …) are the classical Jaboci polynomials, then the system of polynomials of two variables {Ψ mn α,β (x, y)} m,n=0 r = {P m α,β (x)P n α,β (y)} m, n=0 r (r = m + n ≤ N − 1) is an orthogonal system on the set Ω N×N = ⨑ub;(x i , y i ) i,j=0 N , where x i and y i are the zeros of the Jacobi polynomial P n α,β (x). Given an arbitrary continuous function f(x, y) on the square [−1, 1]2, we construct the discrete partial Fourier-Jacobi sums of the rectangular type S m, n, N α,β (f; x, y) by the orthogonal system introduced above. We prove that the order of the Lebesgue constants ∥S m, n, N α,β ∥ of the discrete sums S m, n, N α,β (f; x, y) for −1/2 < α, β < 1/2, m + n ≤ N − 1 is O((mn) q + 1/2), where q = max⨑ub;α,β⫂ub;. As a consequence of this result, several approximate properties of the discrete sums S m, n, N α,β (f; x, y) are considered.

Lebesgue constants of Gaussian cardinal interpolation operators from l (ℤ) into L (ℝ)

In recent years, with the attention to the radial-basis function by mathematicians, more and more research is concentrated on the Gaussian cardinal interpolation. The main purpose of this paper is to discuss the asymptotic behavior of Lebesgue constants of the Gaussian cardinal interpolation operator ℒλ from l (ℤ) into L (ℝ), that is, ∥ℒλ∥1. We obtain the strong asymptotic estimate of the Lebesgue constants which improves the results of Riemenschneider and Sivakumar in [11].

Hyperinterpolation on the square

We show that hyperinterpolation at (near) minimal cubature points for the product Chebyshev measure, along with Xu compact formula for the corresponding reproducing kernel, provide a simple and powerful polynomial approximation formula in the uniform norm on the square. The Lebesgue constant of the hyperinterpolation operator grows like log 2 of the degree, as that of quasi-optimal interpolation sets recently proposed in the literature. Moreover, we give an accurate implementation of the hyperinterpolation formula with linear cost in the number of cubature points, and we compare it with interpolation formulas at the same set of points.

A comparison of interpolation grids over the triangle or the tetrahedron

A simple strategy for constructing a sequence of increasingly refined interpolation grids over the triangle or the tetrahedron is discussed with the goal of achieving uniform convergence and ensuring high interpolation accuracy. The interpolation nodes are generated based on a one-dimensional master grid comprised of the zeros of the Lobatto, Legendre, Chebyshev, and second-kind Chebyshev polynomials. Numerical computations show that the Lebesgue constant and interpolation accuracy of some proposed grids compare favorably with those of alternative grids constructed by optimization, including the Fekete set. While some sets are clearly preferable to others, no single set can claim uniformly better convergence properties as the number of nodes is raised.

ON THE ESTIMATION OF GENERALIZED LEBESGUE CONSTANT AND MODULUS OF GENERALIZED SINGULAR QUADRATURE FORMULAS AND ITS APPLICATION

This article, first gives the estimaties of two modulus, namely, generalized Lebesgue constant and modulus of generalized singular integral quadrature formulas, then applies them to obtain the error bounds of the operator B L m p to the operator B.

An explicit construction of interpolation nodes on the simplex

An open question concerns the spatial distribution of nodes that are suitable for high-order Lagrange interpolation on the triangle and tetrahedron. Several current methods used to produce nodal sets with small Lebesgue constants are recalled. A new approach is presented for building nodal distributions of arbitrary order, that is based on curvilinear finite-element techniques. Numerical results are shown which demonstrate that, despite the explicit nature of this construction, the resulting node sets are well suited for interpolation and competitive with existing sets for up to tenth-order polynomial interpolation. Matlab scripts which evaluate the node distributions on the equilateral triangle are included.

On the behavior of disk algebra bases with applications

The present paper was motivated by an article [H. Akcay, On the existence of a disk algebra basis, Signal Processing 80 (2000) 903–907] on a basis in the disk algebra. Such bases play a central role for the representation of linear systems. In this article it was shown that the Lebesgue constant of a certain set of rational orthogonal functions in the disk algebra diverges. The present paper provides a generalization of this result. It shows that for any arbitrary complete orthonormal set of functions in the disk algebra the Lebesgue constant diverges. However, even if the Lebesgue constant diverges the orthonormal set may still be a disk algebra basis. Moreover, the paper discusses some implications of the divergence result with regard to the robustness of basis representations.

Bivariate Lagrange interpolation at the Padua points: The generating curve approach

We give a simple, geometric and explicit construction of bivariate interpolation at certain points in a square (called Padua points), giving compact formulas for their fundamental Lagrange polynomials. We show that the associated norms of the interpolation operator, i.e., the Lebesgue constants, have minimal order of growth of O ( ( log n ) 2 ) . To the best of our knowledge this is the first complete, explicit example of near optimal bivariate interpolation points.

On group Fourier analysis and symmetry preserving discretizations of PDEs

In this paper we review some group theoretic techniques applied to discretizations of PDEs. Inspired by the recent years active research in Lie group- and exponential-time integrators for differential equations, we will in the first part of the paper present algorithms for computing matrix exponentials based on Fourier transforms on finite groups. As an example, we consider spherically symmetric PDEs, where the discretization preserves the 120 symmetries of the icosahedral group. This motivates the study of spectral element discretizations based on triangular subdivisions. In the second part of the paper, we introduce novel applications of multivariate non-separable Chebyshev polynomials in the construction of spectral element bases on triangular and simplicial sub-domains. These generalized Chebyshev polynomials are intimately connected to the theory of root systems and Weyl groups (used in the classification of semi-simple Lie algebras), and these polynomials share most of the remarkable properties of the classical Chebyshev polynomials, such as near-optimal Lebesgue constants for the interpolation error, the existence of FFT-based algorithms for computing interpolants and pseudo-spectral differentiation and existence of Gaussian integration rules. The two parts of the paper can be read independently.

On the Lebesgue constant for the Xu interpolation formula

In the paper [Y. Xu, Lagrange interpolation on Chebyshev points of two variables, J. Approx. Theory 87 (1996) 220–238], the author introduced a set of Chebyshev-like points for polynomial interpolation (by a certain subspace of polynomials) in the square [ - 1 , 1 ] 2 , and derived a compact form of the corresponding Lagrange interpolation formula. In [L. Bos, M. Caliari, S. De Marchi, M. Vianello, A numerical study of the Xu polynomial interpolation formula in two variables, Computing 76(3–4) (2005) 311–324], we gave an efficient implementation of the Xu interpolation formula and we studied numerically its Lebesgue constant, giving evidence that it grows like O ( ( log n ) 2 ) , n being the degree. The aim of the present paper is to provide an analytic proof to show that the Lebesgue constant does have this order of growth.

A Lobatto interpolation grid in the tetrahedron

A sequence of increasingly refined interpolation grids inside the tetrahedron is proposed with the goal of achieving uniform convergence and ensuring high interpolation accuracy. The number of interpolation nodes, N, corresponds to the number of terms in the complete mth-order polynomial expansion with respect to the three tetrahedral barycentric coordinates. The proposed grid is constructed by deploying Lobatto interpolation nodes over the faces of the tetrahedron, and then computing interior nodes using a simple formula that involves the zeros of the Lobatto polynomials. Numerical computations show that the Lebesgue constant and interpolation accuracy of the proposed grid compare favourably with those of alternative grids constructed by solving optimization problems. The condition number of the mass matrix is significantly lower than that of the uniform grid and comparable to that of optimal grids proposed by previous authors.

A Lobatto interpolation grid over the triangle

A sequence of increasingly refined interpolation grids over the triangle is proposed, with the goal of achieving uniform convergence and ensuring high interpolation accuracy. The number of interpolation nodes, N , corresponds to a complete mth-order polynomial expansion with respect to the triangle barycentric coordinates, which arises by the horizontal truncation of the Pascal triangle. The proposed grid is generated by deploying Lobatto interpolation nodes along the three edges of the triangle, and then computing interior nodes by averaged intersections to achieve three-fold rotational symmetry. Numerical computations show that the Lebesgue constant and interpolation accuracy of the proposed grid compares favorably with those of the best-known grids consisting of the Fekete points. Integration weights corresponding to the set of Lobatto triangle base points are tabulated.

Partitions of a Simplex Leading to Accurate Spectral (Finite) Volume Reconstruction

In this paper, we present several systematic techniques, based on the Voronoi diagram and its variants, to partition a one- and two-dimensional simplex. The Fekete points are used as input to generate the Voronoi diagram, as they concentrate near the edges and are almost optimal for polynomial interpolation in a simplex. Spectral (finite) volume reconstructions on the resultant partitions have small Lebesgue constants. When using the Dubiner basis, the reconstruction matrix is well conditioned. Moreover, the total number of edges of the partitions (the total work when being used in spectral volume methods) is shown to be at most twice the minimum number of edges of all partitions for reconstructions of the same order accuracy. These suggest that the obtained partitions are well suited for spectral volume methods and other numerical methods which rely on reconstructions from cell averages.

Partitions for Spectral (Finite) Volume Reconstruction in the Tetrahedron

In this paper, we compute partitions of the tetrahedron for up to the fourth-order spectral volume reconstruction. Certain optimization is made to these partitions and previously obtained partitions of lower dimensional simplex. These optimized partitions have the smallest Lebesgue constants among currently known spectral volume partitions.

Hermite and Hermite–Fejér interpolation for Stieltjes polynomials

Let w λ ( x ) := ( 1 − x 2 ) λ − 1 / 2 w_{\lambda }(x):=(1-x^2)^{\lambda -1/2} and P n ( λ ) P_n^{(\lambda )} be the ultraspherical polynomials with respect to w λ ( x ) w_{\lambda }(x) . Then we denote by E n + 1 ( λ ) E_{n+1}^{(\lambda )} the Stieltjes polynomials with respect to w λ ( x ) w_{\lambda }(x) satisfying ∫ − 1 1 w λ ( x ) P n ( λ ) ( x ) E n + 1 ( λ ) ( x ) x m d x { = 0 , a m p ; 0 ≤ m &gt; n + 1 , ≠ 0 , a m p ; m = n + 1. \begin{eqnarray*} \int _{-1}^1 w_{\lambda }(x) P_n^{(\lambda )}(x)E_{n+1}^{(\lambda )}(x) x^m dx \begin {cases} =0, &amp; 0 \le m &gt; n+1,\\ \neq 0, &amp; m=n+1. \end{cases} \end{eqnarray*} In this paper, we show uniform convergence of the Hermite–Fejér interpolation polynomials H n + 1 [ ⋅ ] H_{n+1}[\cdot ] and H 2 n + 1 [ ⋅ ] {\mathcal H}_{2n+1}[\cdot ] based on the zeros of the Stieltjes polynomials E n + 1 ( λ ) E_{n+1}^{(\lambda )} and the product E n + 1 ( λ ) P n ( λ ) E_{n+1}^{(\lambda )}P_n^{(\lambda )} for 0 ≤ λ ≤ 1 0 \le \lambda \le 1 and 0 ≤ λ ≤ 1 / 2 0 \le \lambda \le 1/2 , respectively. To prove these results, we prove that the Lebesgue constants of Hermite–Fejér interpolation operators for the Stieltjes polynomials E n + 1 ( λ ) E_{n+1}^{(\lambda )} and the product E n + 1 ( λ ) P n ( λ ) E_{n+1}^{(\lambda )}P_n^{(\lambda )} are optimal, that is, the Lebesgue constants ‖ H n + 1 ‖ ∞ ( 0 ≤ λ ≤ 1 ) \|H_{n+1}\|_{\infty }(0 \le \lambda \le 1) and ‖ H 2 n + 1 ‖ ∞ ( 0 ≤ λ ≤ 1 / 2 ) \|{\mathcal H}_{2n+1}\|_{\infty } (0 \le \lambda \le 1/2) have optimal order O ( 1 ) O(1) . In the case of the Hermite–Fejér interpolation polynomials H 2 n + 1 [ ⋅ ] {\mathcal H}_{2n+1}[\cdot ] for 1 / 2 &gt; λ ≤ 1 1/2 &gt; \lambda \le 1 , we prove weighted uniform convergence. Moreover, we give some convergence theorems of Hermite–Fejér and Hermite interpolation polynomials for 0 ≤ λ ≤ 1 0 \le \lambda \le 1 in weighted L p L_p norms.

A Numerical Study of the Xu Polynomial Interpolation Formula in Two Variables

In his paper "Lagrange interpolation on Chebyshev points of two variables" (J. Approx. Theor. 87, 220–238, 1996), Y. Xu proposed a set of Chebyshev like points for polynomial interpolation in the square [−1,1]2, and derived a compact form of the corresponding Lagrange interpolation formula. We investigate computational aspects of the Xu polynomial interpolation formula like numerical stability and efficiency, the behavior of the Lebesgue constant, and its application to the reconstruction of various test functions.

Explicit form of the Dirichlet kernel for Walsh series and transformations

The dynamics of the behaviour of the absolute values of the Dirichlet kernels is described: the discrete dynamics for the Dirichlet kernel of the Walsh system and the continuous dynamics for the generalized Walsh-Dirichlet kernel. Estimates of the -norms of the Dirichlet kernels are obtained. The concept of generalized Lebesgue constant is introduced and the corresponding formulae are found, which generalize Fine's formulae for the Lebesgue constants. These results hold not only for the Walsh system in the Paley enumeration, but also for rearrangements of the Walsh systems, including linear and piecewise linear ones.

On the Order of the Lebesgue Constants for Interpolation by Algebraic Polynomials from Values at Uniform Nodes of a Simplex

For interpolation processes by algebraic polynomials of degree n from values at uniform nodes of an m-simplex, where m ≥ 2, we obtain the order of growth in n of the Lebesgue constants, which coincides with that in the one-dimensional case for which Turetskii obtained an asymptotics earlier.