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ON EXPONENTIAL CONVERGENCE OF GEGENBAUER INTERPOLATION AND SPECTRAL DIFFERENTIATION
"... Abstract. This paper is devoted to a rigorous analysis of exponential convergence of polynomial interpolation and spectral differentiation based on the GegenbauerGauss and GegenbauerGaussLobatto points, when the underlying function is analytic on and within an ellipse. Sharp error estimates in th ..."
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Abstract. This paper is devoted to a rigorous analysis of exponential convergence of polynomial interpolation and spectral differentiation based on the GegenbauerGauss and GegenbauerGaussLobatto points, when the underlying function is analytic on and within an ellipse. Sharp error estimates
Interpolation
"... mial g of degree n such that (Lgr1) a 2 A ) g(a) = f(a) Denote by R(A ;B ) the product (a \Gamma b), and by A n B the set difference. One remarks that for every a 2 A the polynomial R(x;A n a) vanishes in all the points of A other than a. It is clear that by linear combination of these n + 1 ..."
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mial g of degree n such that (Lgr1) a 2 A ) g(a) = f(a) Denote by R(A ;B ) the product (a \Gamma b), and by A n B the set difference. One remarks that for every a 2 A the polynomial R(x;A n a) vanishes in all the points of A other than a. It is clear that by linear combination of these n + 1
A survey of weighted polynomial approximation with exponential weights
 APPROXIMATION THEORY
, 2007
"... Let W: R! (0, 1] be continuous. Bernstein's approximation problem, posed in 1924,deals with approximation by polynomials in the weighted uniform norm f! kfW kL1(R). Thequalitative form of this problem was solved by Achieser, Mergelyan, and Pollard, in the 1950's. Quantitative forms of the ..."
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of the problem were actively investigated starting from the 1960's. Wesurvey old and recent aspects of this topic, including the Bernstein problem, weighted Jackson and Bernstein Theorems, MarkovBernstein and Nikolskii inequalities, orthogonal expansionsand Lagrange interpolation. We present the main ideas
Weighted convergence of Lagrange interpolation based on GaussKronrod nodes
 J. Comput. Anal. Appl
"... The GaussKronrod quadrature scheme, which is based on the zeros of Legendre polynomials and Stieltjes polynomials, is the standard method for automatic numerical integration in mathematical software libraries. For a long time, very little was known about the underlying Lagrange interpolation proces ..."
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, with additional results that extend and complete the existing ones. The principal new results in this paper are necessary and sufficient conditions for weighted convergence. In particular, we show that the Lagrange interpolation polynomials are equivalent to the polynomials of best approximation in certain
Interpolation via weighted `1 minimization
, 2014
"... Functions of interest are often smooth and sparse in some sense, and both priors should be taken into account when interpolating sampled data. Classical linear interpolation methods are effective under strong regularity assumptions, but cannot incorporate nonlinear sparsity structure. At the same t ..."
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, we provide specific choices of weights in the `1 objective to achieve rates for functions with coefficient sequences in weighted `p spaces, p ≤ 1. We consider the implications of these results for spherical harmonic and polynomial interpolation, in the univariate and multivariate setting. Along
Rational interpolation: II. Quadrature and convergence ⋆
"... Consider an nth rational interpolatory quadrature rule Jσn (f) = ∑n j=1 λjf(xj) to approximate integrals of the form Jσ(f) = ∫ 1 −1 f(x)dσ(x), where σ is a (possibly complex) bounded measure with infinite support on the interval [−1, 1]. First, we discuss the connection of Jσn (f) with certain ratio ..."
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rational interpolatory quadratures on the complex unit circle to approximate integrals of the form ∫ π −π f̊(e iθ)dσ̊(θ). Next, we provide conditions to ensure the convergence of Jσn (f) to Jσ(f) for n tending to infinity. Finally, an upper bound for the error on the nth approximation and an estimate
Exponentially Weighted Moving Averages
"... This paper has been peerreviewed but does not include the final publisher proofcorrections or journal pagination. Citation for the published paper: Frisén, M. and Sonesson, C. Optimal surveillance based on exponentially moving averages. ..."
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This paper has been peerreviewed but does not include the final publisher proofcorrections or journal pagination. Citation for the published paper: Frisén, M. and Sonesson, C. Optimal surveillance based on exponentially moving averages.
Results 1  10
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