Lebesgue Constants Research Articles

Lebesgue constants for polyhedral sets and polynomial interpolation on Lissajous–Chebyshev nodes

To analyze the absolute condition number of multivariate polynomial interpolation on Lissajous–Chebyshev node points, we derive upper and lower bounds for the respective Lebesgue constant. The proof is based on a relation between the Lebesgue constant for the polynomial interpolation problem and the Lebesgue constant linked to the polyhedral partial sums of Fourier series. The magnitude of the obtained bounds is determined by a product of logarithms of the side lengths of the considered polyhedral sets and shows the same behavior as the magnitude of the Lebesgue constant for polynomial interpolation on the tensor product Chebyshev grid.

On uniform Lebesgue constants of local exponential splines with equidistant knots

For a linear differential operator L r of arbitrary order r with constant coefficients and real pairwise different roots of the characteristic polynomial, we study Lebesgue constants (the norms of linear operators from C to C) of local exponential splines corresponding to this operator with a uniform arrangement of knots; such splines were constructed by the authors in earlier papers. In particular, for the third-order operator L 3 = D(D 2 − β 2) (β > 0), we find the exact values of Lebesgue constants for two types of local splines and compare these values with Lebesgue constants of exponential interpolation splines.

TWO-SIDED ESTIMATES OF THE LEBESGUE CONSTANTS WITH RESPECT TO VILENKIN SYSTEMS AND APPLICATIONS

AbstractIn this paper, we derive two-sided estimates of the Lebesgue constants for bounded Vilenkin systems, we also present some applications of importance, e.g., we obtain a characterization for the boundedness of a subsequence of partial sums with respect to Vilenkin–Fourier series of H1 martingales in terms of n's variation. The conditions given in this paper are in a sense necessary and sufficient.

On Lebesgue constants for interpolation points on a quasiconformal arc

Using the theory of quasiconformal mappings, we simplify the proof of the recent result by Taylor and Totik (2010) on the behavior of the Lebesgue constants for interpolation points on a compact set in the complex plane.

Unified treatment of several asymptotic expansions concerning some mathematical constants

Recently various approximation formulas for some mathematical constants have been investigated and presented by many authors. In this paper, we first find that the relationship between the coefficients pj and qj is such thatψ(x∑j=0∞qjx−j)∼ln(x∑j=0∞pjx−j),x→∞,where ψ is the logarithmic derivative of the gamma function (often referred to as psi function) and p0=q0=1. Next, by using this result, we give a unified treatment of several asymptotic expansions concerning the Euler–Mascheroni constant, Landau and Lebesgue constants, Glaisher–Kinkelin constant, and Choi–Srivastava constants.

On the Lebesgue constants of Fourier-Laplace series by Riesz Means

An asymptotic formula for the Lesbegue constant of the Riesz means of Fourier-Laplace series on the sphere obtained in this paper.

Higher-order triangular spectral element method with optimized cubature points for seismic wavefield modeling

The mass-lumped method avoids the cost of inverting the mass matrix and simultaneously maintains spatial accuracy by adopting additional interior integration points, known as cubature points. To date, such points are only known analytically in tensor domains, such as quadrilateral or hexahedral elements. Thus, the diagonal-mass-matrix spectral element method (SEM) in non-tensor domains always relies on numerically computed interpolation points or quadrature points. However, only the cubature points for degrees 1 to 6 are known, which is the reason that we have developed a p-norm-based optimization algorithm to obtain higher-order cubature points. In this way, we obtain and tabulate new cubature points with all positive integration weights for degrees 7 to 9. The dispersion analysis illustrates that the dispersion relation determined from the new optimized cubature points is comparable to that of the mass and stiffness matrices obtained by exact integration. Simultaneously, the Lebesgue constant for the new optimized cubature points indicates its surprisingly good interpolation properties. As a result, such points provide both good interpolation properties and integration accuracy. The Courant–Friedrichs–Lewy (CFL) numbers are tabulated for the conventional Fekete-based triangular spectral element (TSEM), the TSEM with exact integration, and the optimized cubature-based TSEM (OTSEM). A complementary study demonstrates the spectral convergence of the OTSEM. A numerical example conducted on a half-space model demonstrates that the OTSEM improves the accuracy by approximately one order of magnitude compared to the conventional Fekete-based TSEM. In particular, the accuracy of the 7th-order OTSEM is even higher than that of the 14th-order Fekete-based TSEM. Furthermore, the OTSEM produces a result that can compete in accuracy with the quadrilateral SEM (QSEM). The high accuracy of the OTSEM is also tested with a non-flat topography model. In terms of computational efficiency, the OTSEM is more efficient than the Fekete-based TSEM, although it is slightly costlier than the QSEM when a comparable numerical accuracy is required.

The Lebesgue constant of local cubic splines with equally-spaced knots

The Lebesgue constant of local cubic splines with equally-spaced knots

A uniform bound for the Lagrange polynomials of Leja points for the unit disk

We study uniform estimates for the family of fundamental Lagrange polynomials defined from any Leja sequence for the complex unit disk. The main result claims that all these polynomials are uniformly bounded on the disk, i.e. independently on the range $N$ of the associated $N$-Leja section. An essential and immediate application is an estimate of the associated Lebesgue constants as $\Lambda_N=O(N)$, $\forall\,N\geq1$. We also deal with the special case of $N=2^p-1$ where we give an improvement of the main result.

Lebesgue constants arising in a class of collocation methods

Estimates are obtained for the Lebesgue constants associated with the Gauss quadrature points on $(-1, +1)$ augmented by the point $-1$ and with the Radau quadrature points on either $(-1, +1]$ or $[-1, +1)$. It is shown that the Lebesgue constants are $O(\sqrt{N})$, where $N$ is the number of quadrature points. These point sets arise in the estimation of the residual associated with recently developed orthogonal collocation schemes for optimal control problems. For problems with smooth solutions, the estimates for the Lebesgue constants can imply an exponential decay of the residual in the collocated problem as a function of the number of quadrature points.

A sharp bound on the Lebesgue constant for Leja points in the unit disk

We give a sharp bound for the Lebesgue constant associated with Leja sequences in the complex unit disk, confirming a conjecture made by Calvi and Phung in 2011.

Optimal asymptotic Lebesgue constant of Berrut’s rational interpolation operator for equidistant nodes

In approximation theory, the Lebesgue constant of an interpolation operator plays an important role. The Lebesgue constant of Berrut’s interpolation operator has been extensive studied. In the present work, by introducing a new method, we obtain an optimal asymptotic Lebesgue constant of Berrut’s rational interpolant at equidistant nodes.

Lebesgue Constant Using Sinc Points

Lebesgue constant for Lagrange approximation at Sinc points will be examined. We introduce a new barycentric form for Lagrange approximation at Sinc points. Using Thiele’s algorithm we show that the Lebesgue constant grows logarithmically as the number of interpolation Sinc points increases. A comparison between the obtained upper bound of Lebesgue constant using Sinc points and other upper bounds for different set of points, like equidistant and Chebyshev points, is introduced.

The stability of extended Floater–Hormann interpolants

We present a new analysis of the stability of extended Floater---Hormann interpolants, in which both noisy data and rounding errors are considered. Contrary to what is claimed in the current literature, we show that the Lebesgue constant of these interpolants can grow exponentially with the parameters that define them, and we emphasize the importance of using the proper interpretation of the Lebesgue constant in order to estimate correctly the effects of noise and rounding errors. We also present a simple condition that implies the backward instability of the barycentric formula used to implement extended interpolants. Our experiments show that extended interpolants mentioned in the literature satisfy this condition and, therefore, the formula used to implement them is not backward stable. Finally, we explain that the extrapolation step is a significant source of numerical instability for extended interpolants based on extrapolation.

Generalized de la Vallée Poussin approximations on [−1, 1

In this paper, a general approach to de la Vallée Poussin means is given and the resulting near best polynomial approximation is stated by developing simple sufficient conditions to guarantee that the Lebesgue constants are uniformly bounded. Not only the continuous case but also the discrete approximation is investigated and a pointwise estimate of the generalized de Vallée Poussin kernel has been stated to this purpose. The theory is illustrated by several numerical experiments.

Multivariate Lagrange Interpolation at Sinc Points Error Estimation and Lebesgue Constant

This paper gives an explicit construction of multivariate Lagrange interpolation at Sinc points. A nested operator formula for Lagrange interpolation over an $m$-dimensional region is introduced. For the nested Lagrange interpolation, a proof of the upper bound of the error is given showing that the error has an exponentially decaying behavior. For the uniform convergence the growth of the associated norms of the interpolation operator, i.e., the Lebesgue constant has to be taken into consideration. It turns out that this growth is of logarithmic nature $O((log n)^m)$. We compare the obtained Lebesgue constant bound with other well known bounds for Lebesgue constants using different set of points.

Bivariate Lagrange interpolation at the node points of Lissajous curves – the degenerate case

In this article, we study bivariate polynomial interpolation on the node points of degenerate Lissajous figures. These node points form Chebyshev lattices of rank 1 and are generalizations of the well-known Padua points. We show that these node points allow unique interpolation in appropriately defined spaces of polynomials and give explicit formulas for the Lagrange basis polynomials. Further, we prove mean and uniform convergence of the interpolating schemes. For the uniform convergence the growth of the Lebesgue constant has to be taken into consideration. It turns out that this growth is of logarithmic nature.

ON LEBESGUE CONSTANTS OF VILENKIN SYSTEMS

In the paper some properties of Lebesgue constants $\big\{L_n(W)\big\}_{n=1}^{\infty}$ of Vilenkin system are investigated. Non almost convergence property for the sequence $\left\{\dfrac{L_n(W)}{\log_2 n}\right\}_{n=2}^{\infty}$ is obtained.

Lebesgue constants of the Walsh system and Banach limits

We study the properties of the Lebesgue constants of the Walsh system Ln(W), n ∈ N, and apply the results to the theory of Banach limits. We show that the sequence \(\left\{ {\frac{{{L_n}\left( W \right)}}{{{{\log }_2}n}},n \geqslant 2} \right\}\) does not belong to the space of almost convergent sequences ac, which reveals their extremely irregular behavior. Several results of the opposite nature are obtained for some special means of these constants.

A tighter upper bound on the Lebesgue constant of Berrut’s rational interpolant at equidistant nodes

The Lebesgue constant of interpolation operators plays an important role in approximation theory. Several upper bounds have been achieved on the Lebesgue constant of Berrut’s rational interpolation operator. The aim of this paper is to investigate the Lebesgue constant of Berrut’s rational interpolant at equidistant nodes, and present a tight upper bound.