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.

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An improved upper bound on the Lebesgue constant of Berrut’s rational interpolation operator

Journal of Computational and Applied Mathematics ◽

10.1016/j.cam.2013.06.030 ◽

2014 ◽

Vol 255 ◽

pp. 652-660 ◽

Cited By ~ 3

Author(s):

Ren-Jiang Zhang

Keyword(s):

Upper Bound ◽

Lebesgue Constant ◽

Rational Interpolation ◽

Interpolation Operator

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A link between Lebesgue constants and Hermite-Fejér interpolation

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700003075 ◽

1986 ◽

Vol 33 (2) ◽

pp. 207-218 ◽

Cited By ~ 1

Author(s):

S. J. Goodenough

Keyword(s):

Rate Of Convergence ◽

Lagrange Interpolation ◽

Historical Perspective ◽

Lebesgue Constant ◽

The Other ◽

Lebesgue Constants ◽

Close Link ◽

The One ◽

Image Position ◽

Interpolation Polynomials

A review of the development of estimates for Lebesgue constants associated with Lagrange interpolation on the one hand, and estimates for the rate of convergence of Hermite-Fejér interpolation on the other hand, provides a historical perspective for the following surprising, close link between these apparently diverse concepts. Denoting by Λn (T) the Lebesgue constant of order n and by Δn (T) the maximum interpolation error for functions of class Lip 1 by Hexmite-Fejér interpolation polynomials of degree not exceeding 2n − 1, based on the zeros of the Chebyshev polynomial of first kind, we discover that, for even values of n, Λn(T) = n Δn(T).

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Upper Bound for Lebesgue Constant of Bivariate Lagrange Interpolation Polynomial on the Second Kind Chebyshev Points

Journal of Mathematics ◽

10.1155/2022/5649969 ◽

2022 ◽

Vol 2022 ◽

pp. 1-19

Author(s):

Juan Liu ◽

Laiyi Zhu

Keyword(s):

Upper Bound ◽

Chebyshev Polynomials ◽

Lagrange Interpolation ◽

Lebesgue Constant ◽

Interpolation Polynomial ◽

Growth Order ◽

Lagrange Interpolation Polynomial ◽

Chebyshev Points ◽

The Common ◽

Common Zeros

In the paper, we study the upper bound estimation of the Lebesgue constant of the bivariate Lagrange interpolation polynomial based on the common zeros of product Chebyshev polynomials of the second kind on the square − 1,1 2 . And, we prove that the growth order of the Lebesgue constant is O n + 2 2 . This result is different from the Lebesgue constant of Lagrange interpolation polynomial on the unit disk, the growth order of which is O n . And, it is different from the Lebesgue constant of the Lagrange interpolation polynomial based on the common zeros of product Chebyshev polynomials of the first kind on the square − 1,1 2 , the growth order of which is O ln n 2 .

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Lebesgue Constant Using Sinc Points

Advances in Numerical Analysis ◽

10.1155/2016/6758283 ◽

2016 ◽

Vol 2016 ◽

pp. 1-10 ◽

Cited By ~ 3

Author(s):

Maha Youssef ◽

Hany A. El-Sharkawy ◽

Gerd Baumann

Keyword(s):

Upper Bound ◽

Lebesgue Constant ◽

Upper Bounds ◽

Chebyshev Points ◽

Set Of Points ◽

Barycentric Form

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.

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Exact order of the Lebesgue constants for bivariate Lagrange interpolation at certain node-systems

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.2008.1078 ◽

2009 ◽

Vol 46 (1) ◽

pp. 97-102 ◽

Cited By ~ 2

Author(s):

Biancamaria Vecchia ◽

Giuseppe Mastroianni ◽

Péter Vértesi

Keyword(s):

Lagrange Interpolation ◽

Exact Order ◽

Lebesgue Constants

The authors investigate two sets of nodes for bivariate Lagrange interpolation in [−1, 1] 2 and prove that the order of the corresponding Lebesgue constants is (log n ) 2 .

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Criteria for Embedded Eigenvalues for Discrete Schrödinger Operators

International Mathematics Research Notices ◽

10.1093/imrn/rnz262 ◽

2019 ◽

Cited By ~ 3

Author(s):

Wencai Liu

Keyword(s):

Upper Bound ◽

Schrödinger Operators ◽

Schrodinger Operators ◽

Embedded Eigenvalues ◽

Finite Set ◽

Discrete Schrödinger Operators ◽

Sign Type ◽

Free Operator ◽

Set Of Points ◽

Rational Type

Abstract In this paper, we consider discrete Schrödinger operators of the form, $$\begin{equation*} (Hu)(n) = u({n+1})+u({n-1})+V(n)u(n). \end{equation*}$$We view $H$ as a perturbation of the free operator $H_0$, where $(H_0u)(n)= u({n+1})+u({n-1})$. For $H_0$ (no perturbation), $\sigma _{\textrm{ess}}(H_0)=\sigma _{\textrm{ac}}(H)=[-2,2]$ and $H_0$ does not have eigenvalues embedded into $(-2,2)$. It is an interesting and important problem to identify the perturbation such that the operator $H_0+V$ has one eigenvalue (finitely many eigenvalues or countable eigenvalues) embedded into $(-2,2)$. We introduce the almost sign type potentials and develop the Prüfer transformation to address this problem, which leads to the following five results. 1: We obtain the sharp spectral transition for the existence of irrational type eigenvalues or rational type eigenvalues with even denominators.2: Suppose $\limsup _{n\to \infty } n|V(n)|=a<\infty .$ We obtain a lower/upper bound of $a$ such that $H_0+V$ has one rational type eigenvalue with odd denominator.3: We obtain the asymptotical behavior of embedded eigenvalues around the boundaries of $(-2,2)$.4: Given any finite set of points $\{ E_j\}_{j=1}^N$ in $(-2,2)$ with $0\notin \{ E_j\}_{j=1}^N+\{ E_j\}_{j=1}^N$, we construct the explicit potential $V(n)=\frac{O(1)}{1+|n|}$ such that $H=H_0+V$ has eigenvalues $\{ E_j\}_{j=1}^N$.5: Given any countable set of points $\{ E_j\}$ in $(-2,2)$ with $0\notin \{ E_j\}+\{ E_j\}$, and any function $h(n)>0$ going to infinity arbitrarily slowly, we construct the explicit potential $|V(n)|\leq \frac{h(n)}{1+|n|}$ such that $H=H_0+V$ has eigenvalues $\{ E_j\}$.

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MIXED MULTIFRACTAL ANALYSIS FOR FUNCTIONS: GENERAL UPPER BOUND AND OPTIMAL RESULTS FOR VECTORS OF SELF-SIMILAR OR QUASI-SELF-SIMILAR OF FUNCTIONS AND THEIR SUPERPOSITIONS

Fractals ◽

10.1142/s0218348x16500390 ◽

2016 ◽

Vol 24 (04) ◽

pp. 1650039 ◽

Cited By ~ 1

Author(s):

MOURAD BEN SLIMANE ◽

ANOUAR BEN MABROUK ◽

JAMIL AOUIDI

Keyword(s):

Hausdorff Dimension ◽

Upper Bound ◽

Multifractal Analysis ◽

Hölder Functions ◽

Single Function ◽

Cascade Models ◽

Self Similar ◽

Wavelet Decompositions ◽

Vector Formula ◽

Set Of Points

Mixed multifractal analysis for functions studies the Hölder pointwise behavior of more than one single function. For a vector [Formula: see text] of [Formula: see text] functions, with [Formula: see text], we are interested in the mixed Hölder spectrum, which is the Hausdorff dimension of the set of points for which each function [Formula: see text] has exactly a given value [Formula: see text] of pointwise Hölder regularity. We will conjecture a formula which relates the mixed Hölder spectrum to some mixed averaged wavelet quantities of [Formula: see text]. We will prove an upper bound valid for any vector of uniform Hölder functions. Then we will prove the validity of the conjecture for self-similar vectors of functions, quasi-self-similar vectors and their superpositions. These functions are written as the superposition of similar structures at different scales, reminiscent of some possible modelization of turbulence or cascade models. Their expressions look also like wavelet decompositions.

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