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Topic: Lebesgue constant (interpolation)

	Polynomial interpolation - Wikipedia, the free encyclopedia
		In the mathematical subfield of numerical analysis, polynomial interpolation is the interpolation of a given data set by a polynomial.
		One method is to write the interpolation polynomial in the Newton form and use the method of divided differences to construct the coefficients.
		The Lebesgue constant L is defined as the operator norm of X.
	en.wikipedia.org /wiki/Polynomial_interpolation (1428 words)

	Polynomial interpolation: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-10-11)
		One method is to write the interpolation polynomial in the Newton form In the mathematical subfield of numerical analysis, a newton polynomial, named after its inventor isaac newton, is the interpolation polynomial for a given set of data points in the newton form....
		The Lebesgue constant L is defined as the operator norm In mathematics, the operator norm is a means to measure the "size" of certain linear operators....
		Trigonometric interpolation In the mathematical subfield of numerical analysis, trigonometric interpolation is a special form of interpolation on the unit circle in the complex plane using trigonometric polynomials....
	www.absoluteastronomy.com /p/polynomial_interpolation (2488 words)

	Research Experience for Undergraduates
		Behaviour of the BMO-Norm of the Lagrange Interpolating Polynomial
		On the integral of fundamental polynomials of Lagrange interpolation.
		The optimal interval of interpolation by Lagrange polynomials.
	math.fullerton.edu /mathews/n2003/lagrangepoly/LagrangePolyBib/Links/LagrangePolyBib_lnk_3.html (1347 words)

	Polynomial Interpolation
		An important part of the research has involved studies of the Lebesgue function and Lebesgue constant, which are of crucial importance in determining the convergence behaviour of the (0,1,.....,m) HF interpolation polynomials as the number n of nodes increases.
		The graphs also suggest that for fixed m, the Lebesgue constant for (0,1,.....,2m) HF interpolation on the Chebyshev nodes is an increasing function of n, the number of nodes.
		Lebesgue Constants in Polynomial Interpolation, presented to the Department of Mathematics, University of Salzburg, Austria, October 2004
	www.latrobe.edu.au /maths/smith/interp.html (708 words)

	Encyclopedia :: encyclopedia : Mathematical analysis (Site not responding. Last check: 2007-10-11)
		In the early 20th century, calculus was formalized using axiomatic set theory.
		Lebesgue solved the problem of measure, and Hilbert introduced Hilbert spaces to solve integral equations.
		The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis.
	www.hallencyclopedia.com /Mathematical_analysis (692 words)

	[No title]
		Interpolation of data of power growth; \JAT; 6; 1972; 404--420; %Schoenberg72b \rhl{S} \refR Schoenberg, I. Notes on spline functions II.
		Interpolation and curve fitting by sectionally linear functions; \CMB; 3; 1960; 41--57; %Schwerdtfeger61 \rhl{S} \refJ Schwerdtfeger, H.; Notes on numerical analysis III.
		N.; Error of the approximation by interpolation polynomials of small degrees on $n$-simplices (Russian); \MaZ; 48(4); 1990; 88--98; %Subbotin90c % shayne 20feb96 \rhl{S} \refJ Subbotin, Yu.
	www.focm.net /at/spline/bib/S (2647 words)

	PlanetMath:
		Lebesgue number (in Lebesgue number lemma) owned by djao
		Lebesgue's dominated convergence theorem (=dominated convergence theorem) owned by Koro
		Lebesgue's monotone convergence theorem (=monotone convergence theorem) owned by Koro
	planetmath.org /encyclopedia/L (2073 words)

	[No title]
		It is shown that the rate of convergence of the interpolative polynomials depends on the choice of the sequence of multiindices and, for some sequences, is equal to the rate of the best approximation of the interpolated function.
		In the paper [1] the collocation method for singular integral equations and periodic pseudodifferential equations in 1-dimensional Sobolev space was justified.
		estimation of the Lebesgue constant) of the Lagrange interpolation operator in multidimensional Sobolev spaces is needed.
	www.maths.tcd.ie /EMIS/journals/LJM/vol14/fed-mml.html (322 words)

	Publications on Polynomial Interpolation
		Below is a list of publications (since 1990) in the area of polynomial interpolation by members of the Department of Mathematics and Statistics at the Bendigo campus of La Trobe University.
		Simon J. Smith, Lebesgue constants in polynomial interpolation, Annales Mathematicae et Informaticae (to appear).
		Simon J. Smith, The Lebesgue function for Lagrange interpolation on the augmented Chebyshev nodes, Publ.
	www.latrobe.edu.au /maths/smith/int-publ.html (538 words)

	Function
		Locally constant function In neighborhood U of a, such that f is constant on U. Every constant function is locally const...
		k - the constant is 0.33-0.45 in infants, 0.55 in children or adolescent girls, or 0.70 in...
		Sinc function The sinc function, also known as the interpolation function or filtering function, is the product of a m...
	www.brainyencyclopedia.com /topics/function.html (2873 words)

	The Future of Mathematical Text: Mayans: JoDI
		But the fundamental change to come in mathematical publication is not just moving print forms to electronic documents, but recreating mathematics in a new architecture: a hypertext that reflects the deep unity and universality of mathematics, that can grow and diversify as mathematics changes.
		It is argued that hypertext is a natural representation of mathematical thought, with its deep interconnection of ideas, the need for constant revision, and the multiplicity of viewpoints.
		Mathematics is in a constant state of flux.
	jodi.tamu.edu /Articles/v05/i01/Mayans (10772 words)

	INTERPOLATION OF FUNCTIONS (Site not responding. Last check: 2007-10-11)
		This book gives a systematic survey on the most significant results of interpolation theory in the last forty years.
		It deals with Lagrange interpolation including lower estimates, fine and rough theory, interpolatory proofs of Jackson and Teliakovski-Gopengauz theorems, Lebesgue function, Lebesgue constant of Lagrange interpolation, Bernstein and Erdös conjecture on the optimal nodes, the almost everywhere divergence of Lagrange interpolation for arbitrary system of nodes, Hermite-Fejer type and lacunary interpolation and other related topics.
		this is a very careful and exhaustive treatment of interpolation by univariate polynomials...
	www.worldscibooks.com /mathematics/0861.htm (151 words)

	Department of Mathematics and Statistics, UMBC (Site not responding. Last check: 2007-10-11)
		Almost optimal polynomial interpolation points are used as the input.
		The resultant partitions are very accurate (have small Lebesgue constants), the number of edges (roughly proportional to computational cost) for 2D partitions is shown to at most twice the minimum number of edges for the same order reconstruction.
		The optimized partitions have the smallest Lebesgue constant among currently available partitions.
	www.math.umbc.edu /~muruhan/MathColl/feb13_06.htm (152 words)

	[No title] (Site not responding. Last check: 2007-10-11)
		Write the n+1 Chebyshev interpolation points on the interval [-2,2].
		(b) Is the interpolation by cubic splines a linear operator or not?
		Consider the Bernstein interpolation operator B_n that produces polynomials of the degree not larger than n.
	www.math.biu.ac.il /~krasnov/532/exam.html (285 words)

	[No title] (Site not responding. Last check: 2007-10-11)
		S.B. Damelin, The Lebesgue constant of Lagrange interpolation for Erdos weights, J. Approx.
		S.B. Damelin, The weighted Lebesgue constant of Lagrange interpolation for exponential weights on [-1,1], Acta-Mathematica (Hungarica)., 81(3) (1998), pp 211-228.
		S.B. Damelin, H.S. Jung and K.H. Kwon, A note on mean convergence of Lagrange interpolation in Lp, Journal of Computational and Applied mathematics, 133 (1-2) (2001), pp 277-282.
	www.ima.umn.edu /~damelin/paperswa (717 words)

	IngentaConnect A Lobatto interpolation grid over the triangle (Site not responding. Last check: 2007-10-11)
		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 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.
	www.ingentaconnect.com /content/oup/imamat/2006/00000071/00000001/art00153 (205 words)

	Bibliography
		J.-P. Berrut and H. Mittelmann, Lebesgue constant minimizing linear rational interpolation of continuous functions over the interval, Computers Math.
		Bulirsch and H. Rutishauser, Interpolation und genäherte Quadratur, Sauer R., Szabó I., Hsg., Mathematische Hilfsmittel des Ingenieurs, Grundlehren der math.
		Graves-Morris, Efficient reliable rational interpolation, in: Padé Approximation and its Applications, Amsterdam, 1980 M. de Gruin and H. van Rossum, eds., LNM 888, Springer-Verlag, Berlin-Heidelberg-New York, 1981, pp.
	plato.la.asu.edu /papers/paper86/node6.html (317 words)

	[No title] (Site not responding. Last check: 2007-10-11)
		S.B. Damelin, ``The Lebesgue Constant of Lagrange Interpolation for Erd\Hos Weights'', to appear in J.\ Approx.\ Theory.
		S.B. Damelin, ``The Weighted Lebesgue Constant of Lagrange Interpolation for Exponential weights on $[-1,1]$'', to appear in Acta.\ Mathematica, Hungarica.
		S.B. Damelin, ``Lagrange Interpolation for non Szeg\"o weights on $[-1,1]$'', to appear in the Proceedings of the International Workshop on Approximation Theory and Numerical Analysis, Dedicated to Prof.
	www.wits.ac.za /science/number_theory/97repmem.htm (2479 words)

	[No title] (Site not responding. Last check: 2007-10-11)
		Abstract Rational interpolation by a quotient of two n-th degreee polynomials becomes a linear process if the denominator polynomial is fixed.
		We suggest to determine it such that the corresponding Lebesgue constant is minimized.
		The resulting interpolation scheme shows somewhat unsatisfactory behavior for increasing degree.
	plato.la.asu.edu /abstracts/mittelmann.html (56 words)

	SISC Volume 27 Issue 4
		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.
		These suggest that the obtained partitions are well suited for spectral volume methods and other numerical methods which rely on reconstructions from cell averages.
	epubs.siam.org /SISC/volume-27/art_60138.html (232 words)

	The numerical stability of barycentric Lagrange interpolation -- Higham 24 (4): 547 -- IMA Journal of Numerical Analysis (Site not responding. Last check: 2007-10-11)
		The numerical stability of barycentric Lagrange interpolation -- Higham 24 (4): 547 -- IMA Journal of Numerical Analysis
		Articles by Higham, N. The numerical stability of barycentric Lagrange interpolation
		Lagrange formula only for a poor choice of interpolating points.
	imanum.oxfordjournals.org /cgi/content/abstract/24/4/547 (163 words)

	PrintThisPage (Site not responding. Last check: 2007-10-11)
		There are two kinds of Lebesgue constants, those associated with trigonometric Fourier series and those associated with Lagrange interpolation.
		Both are, however, topics of study in the approximation of functions.
		The values of the first several Lebesgue constants are
	www.mathsoft.com /printThisPage.aspx?1078 (146 words)

	Selected Papers of V.K.Dzyadyk
		Dzyadik, V. K.; Shevchuk, I. Remarks on the Lebesgue constant of the Rogosinski kernel.
		Dzjadik, V. Ivanov, V. On asymptotics and estimates for the uniform norms of the Lagrange interpolation polynomials corresponding to the Chebyshev nodal points.
		Dzjadyk, V. jadyk, S. Ju.; Prypik, A. On the asymptotic behavior of the Lebesgue constant in trigonometric interpolation.
	www.imath.kiev.ua /~funct/dzyadyk/papers.html (1195 words)

	Preprints
		We show that there is initial data that exists in every Triebel-Lizorkin or Besov space (and hence in every Lebesgue and Sobolev space), such that after a finite time, the solution is in no Triebel-Lizorkin or Besov space (and hence in no Lebesgue or Sobolev space).
		We obtain an approximate formula for the distribution of the random variable ∥X∥ in terms of its mean and a certain quantity derived from the K-functional of interpolation theory.
		Proceedings of Conference on Geometry of Spaces at Strobl, Ed: P.F.X. Müller and W. Schachermayer, L.M.S. We show that the constants in Pisier's factorization theorem for (p,1)-summing operators from C(Ω) cannot be improved.
	www.math.missouri.edu /~stephen/preprints (4755 words)

	[No title] (Site not responding. Last check: 2007-10-11)
		(6) S.B. Damelin, The Lebesgue constant of Lagrange interpolation for Erd\H{o}s weights, J. Approx.
		(8) S.B. Damelin, The weighted Lebesgue constant of Lagrange interpolation for exponential weights on $[-1,1]$, Acta-Mathematica (Hungarica)., 81(3) (1998), pp 211-228.
		(21) S.B. Damelin, H.S. Jung and K.H. Kwon, A note on mean convergence of Lagrange interpolation in Lp, Journal of Computational and Applied mathematics, 133 (1-2) (2001), pp 277-282.
	www.cs.georgiasouthern.edu /~damelin/damelinlist.html (1213 words)

	[No title]
		Remark 1.10 With respect to the Lebesgue inequality, it is suggestive to look for the best linear projection from X to A, i.e.
		By considering the linear interpolation EMBED Equation.3 at the endpoints of [a; b] show that, for the linear projectors, both inequalities EMBED Equation.3 may become equalities.
		Prove that, on EMBED Equation.3 the Lebesgue function L coincides with the polynomial EMBED Equation.3 such that EMBED Equation.3 all i, EMBED Equation.3 , EMBED Equation.3 for EMBED Equation.3 .
	www.math.unm.edu /~vageli/courses/Ma313/01.doc (1249 words)

	[No title]
		However, there is a theorem by Lebesgue which allows us to bound the interpolation error in terms of the error one would attain by choosing the best possible polynomial approximation.ª‘óm ¨Theorem (Lebesgue)ª ¨ÏAssume that and we consider a set of M interpolation points then: i.e.
		The code will output the approximate Lebesgue constant as the nodes move.
		The approximate Lebesgue number is: 4.92707087069006¡ª[ó”5ó•7¨Surface Mass Matrices¨¾Recall the DG scheme: We now have defined the set of nodes which we can build the hn Lagrange interpolating polynomials.
	www.caam.rice.edu /~timwar/MA578S03/MA578S03Lecture23a.ppt (960 words)

	Mathematics of Computation
		Applications of optimally local interpolation to interpolatory approximants and compactly supported wavelets
		C. Chui, Construction and applications of interpolation formulas, Multivariate Approximation and Interpolation (W. Haussmann and K. Jetter, eds.), Internat.
		J. De Villiers, A convergence result in nodal spline interpolation, J. Approx.
	www.ams.org /mcom/1996-65-213/S0025-5718-96-00672-2 (468 words)

	The Wavelet Digest, Volume 7, Issue 8 (August 24, 1998)
		On the one hand, we have discovered and shown that some well-known wavelets, the Cohen--Daubechies--Feauveau biorthogonal wavelets, are in fact, the derivatives of a certain family of functions (called fundamental) obtained by the iterative interpolation scheme.
		It was therefore natural to try to build multidimensional wavelets from some iterative interpolation schemes and, in order to apply them, it was necessary to be able to work, for example, on rectangular regions in the plane.
		Damelin The Lebesgue function and Lebesgue constant of Lagrange interpolation for Erd\H{o}s weights 235--262 S.
	cm.bell-labs.com /cm/ms/what/wavelet/digest_07/digest_07.08.html (5433 words)

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