Lebesgue Constants Research Articles

On Optimal Approximations of the Norm of the Fourier Operator by a Family of Logarithmic Functions

The Lebesgue constant corresponding to the classical Fourier operator is approximated by a family of logarithmic functions depending on two parameters. We find optimal values of parameters for which the best uniform approximation of the Lebesgue constant by a specific function of this family is achieved. The case where the corresponding remainder strictly increases is also considered.

On Integral Lebesgue Constants of Local Splines with Uniform Knots

We study the stability properties of generalized local splines of the form $$S(x) = S(f,x) = \sum\limits_{j \in \mathbb{Z}} {{y_j}{B_\varphi }\left( {x + \frac{{3h}}{2} - jh} \right),}\;\;\; x \in \mathbb{R},$$ where ϕ ∈ C1[−h, h] for h > 0, ϕ(0) = ϕ′(0) = 0, ϕ(−x) = ϕ(x)for x ∈ [0; h], ϕ(x) is nondecreasing on [0; h], f is an arbitrary function from ℝ to ℝ, yj = f(jh) for j ∈ ℤ, and $${B_\varphi }(x) = m(h)\left\{ {\begin{array}{*{20}{c}} {\varphi (x),}&{x \in [0;h],} \\ {2\varphi (h) - \varphi (x - h) - \varphi (2h - x),}&{x \in [h;2h],} \\ {\varphi (3h - x),}&{x \in [2h;3h],} \\ {0,}&{x \notin [0;3h],} \end{array}} \right.$$ with m(h) > 0. Such splines were constructed by the author earlier. In the present paper, we calculate the exact values of their integral Lebesgue constants (the norms of linear operators from l to L) on the axis ℝ and on any segment of the axis for a certain choice of the boundary conditions and the normalizing factor m(h) of the spline S.

Analysis of a new class of rational RBF expansions

AbstractWe propose a new method, namely an eigen-rational kernel-based scheme, for multivariate interpolation via mesh-free methods. It consists of a fractional radial basis function (RBF) expansion, with the denominator depending on the eigenvector associated to the largest eigenvalue of the kernel matrix. Classical bounds in terms of Lebesgue constants and convergence rates with respect to the mesh size of the eigen-rational interpolant are indeed comparable with those of classical kernel-based methods. However, the proposed approach takes advantage of rescaling the classical RBF expansion providing more robust approximations. Theoretical analysis, numerical experiments and applications support our findings.

Uniform Lebesgue Constants of Local Spline Approximation

Let a function ϕ ∈ C1[−h, h] be such that ϕ(0) = ϕ'(0) = 0, ϕ(−x) = ϕ(x) for x ∈ [0; h], and ϕ(x) is nondecreasing on [0; h]. For any function f: ℝ → ℝ, we consider local splines of the form $$S(x) = {S_\varphi }(f,x) = \sum\limits_{j \in mathbb{Z}} {{y_i}{B_\varphi }(x + \frac{{3h}}{2} - jh)(x \in \mathbb{R}),} $$ where yj = f(jh), m(h) > 0, and $$B_{\varphi}(x)=m(h) \begin{cases}\varphi(x), & x \in [0;h],\\ 2\varphi (h) - \varphi(x-h) - \varphi(2h-x),& x \in [h;2h],\\ \varphi(3h -x), & x\in[2h;3h],\\ 0, x 3h].\end{cases}$$ These splines become parabolic, exponential, trigonometric, etc., under the corresponding choice of the function ϕ. We study the uniform Lebesgue constants Lϕ = ||S|| C C (the norms of linear operators from C to C) of these splines as functions depending on ϕ and h. In some cases, the constants are calculated exactly on the axis ℝ and on a closed interval of the real line (under a certain choice of boundary conditions from the spline Sϕ(f, x)).

Uniform approximation on the sphere by least squares polynomials

The paper concerns the uniform polynomial approximation of a function f, continuous on the unit Euclidean sphere of ℝ3 and known only at a finite number of points that are somehow uniformly distributed on the sphere. First, we focus on least squares polynomial approximation and prove that the related Lebesgue constants w.r.t. the uniform norm grow at the optimal rate. Then, we consider delayed arithmetic means of least squares polynomials whose degrees vary from n − m up to n + m, being m = θn for any fixed parameter 0 < θ < 1. As n tends to infinity, we prove that these polynomials uniformly converge to f at the near-best polynomial approximation rate. Moreover, for fixed n, by using the same data points, we can further improve the approximation by suitably modulating the action ray m determined by the parameter θ. Some numerical experiments are given to illustrate the theoretical results.

About the Optimal Replacement of the Lebesque Constant Fourier Operator by a Logarithmic Function

The Lebesgue constant, corresponding to the classical Fourier operator is approaching a two-parameter family of logarithmic functions. The optimal values of these parameters are found in which the best uniform approximation of the Lebesgue constant a well-defined function of this family is achieved. We considered the case when corresponding to these functions the remaining members are strictly decreasing.

On the computation of sets of points with low Lebesgue constant on the unit disk

In this paper of numerical nature, we test the Lebesgue constant of several available point sets on the disk Ω and propose new ones that enjoy low Lebesgue constant. Furthermore we extend some results in Cuyt (2012), analyzing the case of Bos arrays whose radii are nonnegative Gauss–Gegenbauer–Lobatto nodes with exponent α , noticing that the optimal α still allows to achieve point sets on Ω with low Lebesgue constant Λ n for degrees n ≤ 30 . Next we introduce an algorithm that through optimization determines point sets with the best known Lebesgue constants for n ≤ 25 . Finally, we determine theoretically a point set with the best Lebesgue constant for the case n = 1 .

On the Lebesgue constant of weighted Leja points for Lagrange interpolation on unbounded domains

This work focuses on weighted Lagrange interpolation on an unbounded domain, and analyzes the Lebesgue constant for a sequence of weighted Leja points. The standard Leja points are a nested sequence of points defined on a compact subset of the real line, and can be extended to unbounded domains with the introduction of a weight function $w:\mathbb{R}\rightarrow [0,1]$. Due to a simple recursive formulation in one dimension, such abscissas provide a foundation for high-dimensional approximation methods such as sparse grid collocation, deterministic least squares, and compressed sensing. Just as in the unweighted case of interpolation on a compact domain, we use results from potential theory to prove that the Lebesgue constant for the Leja points grows subexponentially with the number of interpolation nodes.

On the Lebesgue constant of barycentric rational Hermite interpolants at equidistant nodes

Barycentric rational Floater–Hormann interpolants compare favourably to classical polynomial interpolants in the case of equidistant nodes, because the Lebesgue constant associated with these interpolants grows logarithmically in this setting, in contrast to the exponential growth experienced by polynomials. In the Hermite setting, in which also the first derivatives of the interpolant are prescribed at the nodes, the same exponential growth has been proven for polynomial interpolants, and the main goal of this paper is to show that much better results can be obtained with a recent generalization of Floater–Hormann interpolants. After summarizing the construction of these barycentric rational Hermite interpolants, we study the behaviour of the corresponding Lebesgue constant and prove that it is bounded from above by a constant. Several numerical examples confirm this result.

Lebesgue constants for Chebyshev thresholding greedy algorithms

We investigate the efficiency of Chebyshev Thresholding Greedy Algorithm (CTGA) for an n-term approximation with respect to general bases in a Banach space. We show that the convergence property of CTGA is better than TGA for non-quasi-greedy bases. Then we determine the exact rate of the Lebesgue constants L_{n}^{mathrm{ch}} for two examples of such bases: the trigonometric system and the summing basis. We also establish the upper estimates for L_{n}^{mathrm{ch}} with respect to general bases in terms of quasi-greedy parameter, democracy parameter and A-property parameter. These estimates do not involve an unconditionality parameter, therefore they are better than those of TGA. In particular, for conditional quasi-greedy bases, a faster convergence rate is obtained.

On Lagrange polynomials and the rate of approximation of planar sets by polynomial Julia sets

We revisit the approximation of nonempty compact planar sets by filled-in Julia sets of polynomials developed in [27] and analyze the rate of approximation. We use slightly modified fundamental Lagrange interpolation polynomials and show that taking certain classes of nodes with subexponential growth of Lebesgue constants improves the approximation rate. To this end we investigate properties of some arrays of points in C. In particular we prove subexponential growth of Lebesgue constants for pseudo Leja sequences with bounded Edrei growth on finite unions of quasiconformal arcs. Finally, for some classes of sets we estimate more precisely the rate of approximation by filled-in Julia sets in Hausdorff and Klimek metrics.

On the growth of Lebesgue constants for convex polyhedra

In this paper, new estimates of the Lebesgue constant L ( W ) = 1 ( 2 π ) d ∫ T d | ∑ k ∈ W ∩ Z d e i ( k , x ) | d x \begin{equation*} \mathcal {L}(W)=\frac 1{(2\pi )^d}\int _{{\Bbb T}^d}\bigg |\sum _{\mathbfit {k}\in W\cap {\Bbb Z}^d} e^{i(\mathbfit {k}, \mathbfit {x})}\bigg | \textrm {d}\mathbfit {x} \end{equation*} for convex polyhedra W ⊂ R d W\subset {\Bbb R}^d are obtained. The main result states that if W W is a convex polyhedron such that [ 0 , m 1 ] × ⋯ × [ 0 , m d ] ⊂ W ⊂ [ 0 , n 1 ] × ⋯ × [ 0 , n d ] [0,m_1]\times \dots \times [0,m_d]\subset W\subset [0,n_1]\times \dots \times [0,n_d] , then c ( d ) ∏ j = 1 d log ⁡ ( m j + 1 ) ≤ L ( W ) ≤ C ( d ) s ∏ j = 1 d log ⁡ ( n j + 1 ) , \begin{equation*} c(d)\prod _{j=1}^d \log (m_j+1)\le \mathcal {L}(W)\le C(d)s\prod _{j=1}^d \log (n_j+1), \end{equation*} where s s is a size of the triangulation of W W .

Lebesgue Constants for Some Interpolating L-Splines

We find exact values for the uniform Lebesgue constants of interpolating L-splines that are bounded on the real axis, have equidistant knots, and correspond to the linear thirdorder differential operator L3(D) = D(D2 + α2) with constant real coefficients, where α > 0. We compare the obtained result with the Lebesgue constants of other L-splines.

The Convergence Problems of Eigenfunction Expansions of Elliptic Differential Operators

In the present research we investigate the problems concerning the almost everywhere convergence of multiple Fourier series summed over the elliptic levels in the classes of Liouville. The sufficient conditions for the almost everywhere convergence problems, which are most difficult problems in Harmonic analysis, are obtained. The methods of approximation by multiple Fourier series summed over elliptic curves are applied to obtain suitable estimations for the maximal operator of the spectral decompositions. Obtaining of such estimations involves very complicated calculations which depends on the functional structure of the classes of functions. The main idea on the proving the almost everywhere convergence of the eigenfunction expansions in the interpolation spaces is estimation of the maximal operator of the partial sums in the boundary classes and application of the interpolation Theorem of the family of linear operators. In the present work the maximal operator of the elliptic partial sums are estimated in the interpolation classes of Liouville and the almost everywhere convergence of the multiple Fourier series by elliptic summation methods are established. The considering multiple Fourier series as an eigenfunction expansions of the differential operators helps to translate the functional properties (for example smoothness) of the Liouville classes into Fourier coefficients of the functions which being expanded into such expansions. The sufficient conditions for convergence of the multiple Fourier series of functions from Liouville classes are obtained in terms of the smoothness and dimensions. Such results are highly effective in solving the boundary problems with periodic boundary conditions occurring in the spectral theory of differential operators. The investigations of multiple Fourier series in modern methods of harmonic analysis incorporates the wide use of methods from functional analysis, mathematical physics, modern operator theory and spectral decomposition. New method for the best approximation of the square-integrable function by multiple Fourier series summed over the elliptic levels are established. Using the best approximation, the Lebesgue constant corresponding to the elliptic partial sums is estimated. The latter is applied to obtain an estimation for the maximal operator in the classes of Liouville.

Numerical Integration in Multiple Dimensions with Designed Quadrature

We present a systematic computational framework for generating positive quadrature rules in multiple dimensions on general geometries. A direct moment-matching formulation that enforces exact integration on polynomial subspaces yields nonlinear conditions and geometric constraints on nodes and weights. We use penalty methods to address the geometric constraints, and subsequently solve a quadratic minimization problem via the Gauss-Newton method. Our analysis provides guidance on requisite sizes of quadrature rules for a given polynomial subspace, and furnishes useful user-end stability bounds on error in the quadrature rule in the case when the polynomial moment conditions are violated by a small amount due to, e.g., finite precision limitations or stagnation of the optimization procedure. We present several numerical examples investigating optimal low-degree quadrature rules, Lebesgue constants, and 100-dimensional quadrature. Our capstone examples compare our quadrature approach to popular alternatives, such as sparse grids and quasi-Monte Carlo methods, for problems in linear elasticity and topology optimization.

Multivariate polynomial interpolation using even and odd polynomials

The Lebesgue constant is a measure for the stability of the Lagrange interpolation. The decomposition of the Lagrange interpolation operator in their even and odd parts with respect to the last variable can be used to find a relation between the Lebesgue constant for a space of polynomials and the corresponding Lebesgue constants for subspaces of even and odd polynomials. It is shown that such a decomposition preserves the stability properties of the Lagrange interpolation operator. We use the Lebesgue functions to provide pointwise quantitative measures of the stability properties and illustrate with examples the behaviour in simple cases.

Convergence Analysis for the Chebyshev Collocation Methods to Volterra Integral Equations with a Weakly Singular Kernel

AbstractIn this paper, a Chebyshev-collocation spectral method is developed for Volterra integral equations (VIEs) of second kind with weakly singular kernel. We first change the equation into an equivalent VIE so that the solution of the new equation possesses better regularity. The integral term in the resulting VIE is approximated by Gauss quadrature formulas using the Chebyshev collocation points. The convergence analysis of this method is based on the Lebesgue constant for the Lagrange interpolation polynomials, approximation theory for orthogonal polynomials, and the operator theory. The spectral rate of convergence for the proposed method is established in theL∞-norm and weightedL2-norm. Numerical results are presented to demonstrate the effectiveness of the proposed method.

Lebesgue Constant of Local Cubic Splines with Equally Spaced Nodes

It is proved that the uniform Lebesgue constant (the norm of a linear operator from C to C) of local cubic splines with equally spaced nodes, which preserve cubic polynomials, is equal to 11/9.

Boundedness of Lebesgue Constants and Interpolating Faber Bases

Background. We investigate the relationship between the boundedness of Lebesgue constants for the Lagrange polynomial interpolation on a compact subset of \[\mathbb R\] and the existence of a Faber basis in the space of continuous functions on this compact set. Objective. The aim of the paper is to describe the conditions on the matrix of interpolation nodes under which the interpolation of any continuous function coincides with the decomposition of this function in a series on the Faber basis. Methods. The methods of general theory of Schauder bases and the results which describe the convergence of interpolating Lagrange processes are used. Results. The structure of matrices of interpolation nodes which generate the interpolating Faber bases is described. Conclusions. Every interpolating Faber basis is generated by the interpolating Lagrange process with the interpolating matrix of a special kind and bounded Lebesgue constants.

Upper bounds for uniform Lebesgue constants of interpolation periodic sourcewise representable splines

Upper bounds for Lebesgue constants (norms of linear operators from C to C) of interpolation periodic sourcewise representable splines with uniform knots are obtained for a wide class of periodic integrable kernels K.