Positive-Definite Functions, Exponential Sums and the Greedy Algorithm: a Curious Phenomenon

Brown, Louis D.; Steinerberger, Stefan

doi:10.1016/j.jco.2020.101485

Cited by 11 publications

(18 citation statements)

References 21 publications

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“…Clearly, the greedy averaging search algorithm is a realization of the averaging search algorithm and, therefore, Theorem 7.1 and Corollary 7.1 hold for points obtained by the greedy averaging search algorithm. It is an interesting open problem, which is discussed in detail in [5], to understand if the greedy averaging search algorithm can give better error bounds than m −1/2 .…”

Section: Discussionmentioning

confidence: 99%

“…We begin with the definition of the averaging search algorithm. This algorithm and its greedy version were analyzed in the recent paper [5].…”

Section: Averaging Search Algorithm In Numerical Integrationmentioning

confidence: 99%

“…Proof We begin with a simple identity, which was used in [19] in a context of numerical integration (see also [5]). Then,…”

Section: Averaging Search Algorithm In Numerical Integrationmentioning

confidence: 99%

“…We note that the averaging search algorithm is not a greedy type algorithm. The following greedy version of this algorithm has been studied in a very recent paper [5].…”

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confidence: 99%

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Numerical Integration and Discrepancy Under Smoothness Assumption and Without It

Temlyakov

2021

Constr Approx

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The goal of this paper is threefold. First, we present a unified way of formulating numerical integration problems from both approximation theory and discrepancy theory. Second, we show how techniques, developed in approximation theory, work in proving lower bounds for recently developed new type of discrepancy-the smooth discrepancy. Third, we consider numerical integration in classes, for which we do not impose any smoothness assumptions. We illustrate how nonlinear approximation, in particular greedy approximation, allows us to guarantee some rate of decay of errors of numerical integration even in such a general setting with no smoothness assumptions.

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Section: Discussionmentioning

confidence: 99%

“…We begin with the definition of the averaging search algorithm. This algorithm and its greedy version were analyzed in the recent paper [5].…”

Section: Averaging Search Algorithm In Numerical Integrationmentioning

confidence: 99%

“…Proof We begin with a simple identity, which was used in [19] in a context of numerical integration (see also [5]). Then,…”

Section: Averaging Search Algorithm In Numerical Integrationmentioning

confidence: 99%

“…We note that the averaging search algorithm is not a greedy type algorithm. The following greedy version of this algorithm has been studied in a very recent paper [5].…”

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confidence: 99%

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Numerical Integration and Discrepancy Under Smoothness Assumption and Without It

Temlyakov

2021

Constr Approx

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“…where the polynomial is the second Bernoulli polynomial. Brown and the author [16] have investigated this problem. Numerics suggest that there is very little difference in the behavior of the sequences: the sequence…”

Section: At the Same This Mysterious Function Log (2| Sin (πX)|) Arises Naturally As An Infinite Fourier Seriesmentioning

confidence: 99%

Polynomials With Zeros on the Unit Circle: Regularity of Leja Sequences

Steinerberger

2021

Mathematika

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Let z 1 , . . . , z m be m distinct complex numbers, normalized to |z k | = 1, and consider the polynomialWe define a sequence of polynomials in a greedy fashion,whereand prove that, independently of the initial polynomial p m , the roots of p N equidistribute in angle at rate at most (log N ) 2 /N. This persists when sometimes adding "adversarial" points by hand. We also obtain sharp rates for an L 2 -version of a problem first raised by Erdős and solved by Beck in L ∞ . §1. Introduction.1.1. Introduction. Let (x n ) ∞ n=1 be a sequence on [0,1]. We define the discrepancy function D N : [0, 1] → [0, 1] associated of the first N elements via

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Sequences of Well-Distributed Vertices on Graphs and Spectral Bounds on Optimal Transport

Brown

2021

J Fourier Anal Appl

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Positive-Definite Functions, Exponential Sums and the Greedy Algorithm: a Curious Phenomenon

Cited by 11 publications

References 21 publications

Numerical Integration and Discrepancy Under Smoothness Assumption and Without It

Numerical Integration and Discrepancy Under Smoothness Assumption and Without It

Polynomials With Zeros on the Unit Circle: Regularity of Leja Sequences

Sequences of Well-Distributed Vertices on Graphs and Spectral Bounds on Optimal Transport

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