Lebesgue Constants Research Articles

Two‐sided estimates of Lebesque constants for Hölder spaces

AbstractA two‐sided estimate of Lebesgue constants is proposed for the Hölder space , constructed from the modulus of continuity of the th order calculated in the symmetric space . Choosing different function spaces , we obtain new criteria for the convergence of classical Fourier series.

De la Vallée Poussin filtered polynomial approximation on the half–line

On the half line, we introduce a new sequence of near-best uniform approximation polynomials, easily computable by the values of the approximated function at a truncated number of Laguerre zeros. Such approximation polynomials come from a discretization of filtered Fourier–Laguerre partial sums, which are filtered using a de la Vallée Poussin (VP) filter. They have the peculiarity of depending on two parameters: a truncation parameter that determines how many of the n Laguerre zeros are considered, and a localization parameter, which determines the range of action of the VP filter we will apply. As n→∞, under simple assumptions on such parameters and the Laguerre exponents of the involved weights, we prove that the new VP filtered approximation polynomials have uniformly bounded Lebesgue constants and uniformly convergence at a near–best approximation rate, for any locally continuous function on the semiaxis.The numerical experiments have validated the theoretical results. In particular, they show a better performance of the proposed VP filtered approximation versus the truncated Lagrange interpolation at the same nodes, especially for functions a.e. very smooth with isolated singularities. In such cases, we see a more localized approximation and a satisfactory reduction of the Gibbs phenomenon.

Lebesgue Constants for Cantor Sets

We evaluate the values of the Lebesgue constants in polynomial interpolation for three types of Cantor sets. In all cases, the sequences of Lebesgue constants are not bounded. This disproves the statement by Mergelyan.

On Lebesgue Constants

Simple formulae for Lebesgue constants arising in the classical Fourier series approximation are obtained. Both even and odd cases are addressed, extending Fejér’s results. Asymptotic formulae are also obtained.

Asymptotic expansion of the Lebesgue constant for even-order Hermite-Fejér interpolation on Chebyshev nodes

An integral representation is obtained for the Lebesgue constant associated with even-order Hermite-Fejér interpolation on Chebyshev nodes, and this result is extended to give a complete asymptotic expansion with a remainder term.

Two-sided estimates of Lebesgue constants for spaces of functions of generalized bounded variation and Hölder spaces

A proof of two-sided estimates of Lebesgue constants for spaces Hω⋂BV(X), where Hω is the Hölder space and BV(X) is the space of functions of bounded variation constructed from the symmetric space of sequences X. Choosing different spaces of sequences as X, we obtained new criteria for the convergence of classical Fourier series.

Chebyshev admissible meshes and Lebesgue constants of complex polynomial projections

We construct admissible polynomial meshes on piecewise polynomial or trigonometric curves of the complex plane, by mapping univariate Chebyshev points. Such meshes can be used for polynomial least-squares, for the extraction of Fekete-like and Leja-like interpolation sets, and also for the evaluation of their Lebesgue constants.

Evaluating Lebesgue constants by Chebyshev polynomial meshes on cube, simplex, and ball

Evaluating Lebesgue constants by Chebyshev polynomial meshes on cube, simplex, and ball

On a two-dimensional analogue of the Lebesgue function for Fourier-Chebyshov sums

This article considers the problem of approximating a function of two variables f(x,y) by Fourier sums over Chebyshev polynomials orthogonal on a discrete grid. The paper shows that if Tα,βn(x, N) (α,β > -1, n = 0,1,2,...) are classical Chebyshev polynomials, orthogonal on a discrete grid, then the system of polynomials of two variables {Zα,βm,n(x, y)}km,n=0 = {Tα,βm(x, N)}km,n=0, where k = m + n ≤ N −1 is orthogonal on the set ΩN × N = {(xi, yj)}i,j=1N - 1. For an arbitrary function f(x,y) continuous on [−1,1]2, partial Fourier Chebyshev sums Sα,βm,n,N(f, x,y) are constructed over the system of polynomials τα,βm,n,N(x,y) orthonormal on the grid ΩN × N = {(xi, yj)}i,j=1N - 1. The task is to estimate the partial sum Sα,βm,n,N(f, x,y) of the Fourier series of the function f(x,y) from the system of polynomials τα,βm,n,N(x,y) from the function itself f ∈ C[-1, 1]2, in the case when(x,y) ∈ [−1,1]×[−1,1], which in turn reduces to the problem of estimating the Lebesgue function Wα,βm,n,N(x,y). The result of the work is a theorem in which it is proved that the order of the Lebesgue constants ‖Sα,βm,n,N‖ of the indicated discrete sums under certain conditions is O((mn)q+1/2), where q = max{α,β}. As a consequence of the obtained result, some approximation properties of discrete sums Sα,βm,n,N(f, x,y) are considered.

Estimating the Lebesgue constant for the Chebyshev distribution of nodes

In this paper an approach to estimation of the Lebesgue constant for the Lagrange interpolation process with nodes in the zeros of Chebyshev polynomials of the first kind is done. Two-sided estimation of this constant is carried out by using the logarithmic derivative of the Euler gamma function and of the Riemann zeta function. The choice of interpolation nodes is due to the fact that with a fixed number of Chebyshev nodes, the Lebesgue constant tends to its minimum value, thus reducing the error of algebraic interpolation and providing less sensitivity to rounding errors. The expressions for the upper and the lower bounds of this constant are represented as finite sums of an asymptotic alternating series. Based on the expressions obtained, these boundaries are calculated depending on the number of nodes of the interpolation process. The error of each of the boundaries’ value is estimated based on the first discarded term in the corresponding asymptotic series. The results of the calculations are presented in tables showing deviations of the Lebesgue constant from its lower and upper estimated bounds. Dependence of the values’ errors on the number of Chebyshev nodes is depicted in these tables as well. It is numerically shown that with an increase in the number of these nodes, the estimation boundaries rapidly get close to each other. The presented results can be used in the theory of interpolation to estimate the norm of the operator matching a function to its interpolation polynomial and to estimate a deviation of the constructed perturbed polynomial from the unperturbed one.

Approximation of the Lebesgue constant of the Fourier operator by a logarithmic-fractional-rational function

The Lebesgue constant of the classical Fourier operator is uniformly approximated by a logarithmic-fractional-rational function depending on three parameters; they are defined using the specific properties of logarithmic and rational approximations. A rigorous study of the corresponding residual term having an indefinite (non-monotonic) behavior has been carried out. The obtained approximation results strengthen the known results by more than two orders of magnitude.

A generalization of Floater–Hormann interpolants

In this paper the interpolating rational functions introduced by Floater and Hormann are generalized leading to a whole new family of rational functions depending on γ, an additional positive integer parameter. For γ=1, the original Floater–Hormann interpolants are obtained. When γ>1 we prove that the new rational functions share a lot of the nice properties of the original Floater–Hormann functions. Indeed, for any configuration of nodes in a compact interval, they have no real poles, interpolate the given data, preserve the polynomials up to a certain fixed degree, and have a barycentric-type representation. Moreover, we estimate the associated Lebesgue constants in terms of the minimum (h∗) and maximum (h) distance between two consecutive nodes. It turns out that, in contrast to the original Floater–Hormann interpolants, for all γ>1 we get uniformly bounded Lebesgue constants in the case of equidistant and quasi-equidistant nodes configurations (i.e., when h∼h∗). For such configurations, as the number of nodes tends to infinity, we prove that the new interpolants (γ>1) uniformly converge to the interpolated function f, for any continuous function f and all γ>1. The same is not ensured by the original FH interpolants (γ=1). Moreover, we provide uniform and pointwise estimates of the approximation error for functions having different degrees of smoothness. Numerical experiments illustrate the theoretical results and show a better error profile for less smooth functions compared to the original Floater–Hormann interpolants.

The upper bound for the Lebesgue constant for Lagrange interpolation in equally spaced points of the triangle

An upper bound for the Lebesgue constant, i.e., the supremum norm of the operator of interpolation of a function in equally spaced points of a triangle by a polynomial of total degree less than or equal to n is obtained. Earlier, the rate of increase of the Lebesgue constants with respect to n for an arbitrary d-dimensional simplex was established by the author. The upper bound proved in this article refines this result for d=2.

On solving some Cauchy singular integral equations by de la Vallée Poussin filtered approximation

The paper deals with the numerical solution of Cauchy Singular Integral Equations based on some non standard polynomial quasi–projection of de la Vallée Poussin type. Such kind of approximation presents several advantages over classical Lagrange interpolation such as the uniform boundedness of the Lebesgue constants, the near–best order of uniform convergence to any continuous function, and a strong reduction of Gibbs phenomenon. These features will be inherited by the proposed numerical method which is stable and convergent, and provides a near-best polynomial approximation of the sought solution by solving a well conditioned linear system. The numerical tests confirm the theoretical error estimates and, in case of functions subject to Gibbs phenomenon, they show a better local approximation compared with analogous Lagrange projection methods.

More properties of (β,γ)-Chebyshev functions and points

Recently, (β,γ)-Chebyshev functions, as well as the corresponding zeros, have been introduced as a generalization of classical Chebyshev polynomials of the first kind and related roots. They consist of a family of orthogonal functions on a subset of [−1,1], which indeed satisfies a three-term recurrence formula. In this paper we present further properties, which are proven to comply with various results about classical orthogonal polynomials. In addition, we prove a conjecture concerning the Lebesgue constant's behavior related to the roots of (β,γ)-Chebyshev functions in the corresponding orthogonality interval.

Estimates of Lebesgue constants for Lagrange interpolation processes by rational functions under mild restrictions to their fixed poles

We estimate the Lebesgue constants for Lagrange interpolation processes on one or several intervals by rational functions with fixed poles. We admit that the poles have finitely many accumulation points on the intervals.To prove it we use an analog of the inverse polynomial image method for rational functions with fixed poles.

Blends have decent numerical properties

A "blend" is a two-point Hermite interpolational polynomial, typically of quite high degree. This note shows that implementing them in a double Horner evaluation scheme has good backward error, and also shows that the Lebesgue constant for a balanced blend or nearly balanced blend on the interval [0,1] is bounded by 2, independently of the grade or degree of the approximation. On [-1,1], which is a more natural interval for comparison, it is of course unbounded, but grows only like 2√(m/π) where 2m+1 is the grade of approximation. I also show that the quadrature schemes for balanced blends amplify errors only by O( ln(m) ).

Order of uniform approximation by polynomial interpolation in the complex plane and beyond

Let X denote a compact set in the complex plane ℂ and zn≔{zn,j}j=0n be a family of points that lie on X. Then the norm ‖Azn‖ of the operator Azn that maps the Banach space C(X) to the space Πn of polynomials of degree ≤n, defined by Aznf=pn, where pn∈Πn is the polynomial that interpolates f∈C(X) at zn, is called the Lebesgue constant of the family zn. While the bulk of this paper is on the study of Lagrange polynomial interpolation at the Fejér points zn∗≔{zn,j∗}j=0n, and their proper adjustment, that lie on an open arc X=γ in ℂ which does not cross itself, the general spirit of our presentation carries over to other settings of polynomial representations of functions on a simple closed piece-wise smooth curve X=Γ, which is the boundary of a Jordan domain D⊂ℂ; and instead of polynomial interpolation, Azn is replaced by the linear operator AFn, that maps the (generalized) Hardy space Hp(D), for p≥1, of analytic functions f in D with non-tangential limit f∗∈Lp(Γ), to the nth partial sum Sn(⋅;f) of the Faber series representation of f in D. The importance of the Lebesgue constant ‖AFn‖ is that if Pn∗∈Πn is the best polynomial approximant of f in Hp(D), with approximation error ϵn(f;Γ)≔‖f−Pn∗(⋅;f)‖Hp, then Sn(⋅;f) can be used to replace Pn∗ with error of approximation ‖f−Sn(⋅;f)‖Hp≤(1+‖AFn‖)ϵn(f;Γ). This consideration, along with the discussion of related problems and results inspired by the work of Prof. J. Korevaar, will be presented in the final section of the present paper.For Lagrange polynomial interpolation on open arcs X=γ in ℂ, it is well-known that the Lebesgue constant for the family of Chebyshev points xn≔{xn,j}j=0n on [−1,1]⊂R has growth order of O(log(n)). The same growth order was shown in Zhong and Zhu (1995) for the Lebesgue constant of the family zn∗∗≔{zn,j∗∗}j=0n of some properly adjusted Fejér points on a rectifiable smooth open arc γ⊂ℂ. On the other hand, in our recent work Chui and Zhong (2021), it was observed that if the smooth open arc γ is replaced by an L-shape arc γ0⊂ℂ consisting of two line segments, numerical experiments suggest that the Marcinkiewicz–Zygmund inequalities are no longer valid for the family of Fejér points zn∗≔{zn,j∗}j=0n on γ, and that the rate of growth for the corresponding Lebesgue constant Lzn∗ is as fast as clog2(n) for some constant c>0.The main objective of the present paper is 3-fold: firstly, it will be shown that for the special case of the L-shape arc γ0 consisting of two line segments of the same length that meet at the angle of π/2, the growth rate of the Lebesgue constant Lzn∗ is at least as fast as O(log2(n)), with limsupLzn∗log2(n)=∞; secondly, the corresponding (modified) Marcinkiewicz–Zygmund inequalities fail to hold; and thirdly, a proper adjustment zn∗∗≔{zn,j∗∗}j=0n of the Fejér points on γ will be described to assure the growth rate of Lzn∗∗ to be exactly O(log2(n)).

On the condition number of Newton interpolation: Chebyshev points and monotonically ordering

We obtain sharp analytic approximations for the condition number ΛN of Newton interpolation for Chebyshev nodes of the first and second kinds. Our results show that ΛN grows faster for Chebyshev nodes than for equally spaced nodes, which contrasts with the classical theory of Lebesgue constants. We also obtain sharp approximations for ΛN for Chebyshev points of the second kind in inverse central order and we show that the condition number of Newton interpolation grows exponentially for all families of monotonically ordered nodes.

The Lebesgue constants for Leja points are subexponential

It is shown that on a set of positive logarithmic capacity the Lebesgue constants associated with Leja points are subexponential. Actually, the same result is verified for the so-called τ-Leja points introduced by Andrievskii and Nazarov.