Optimal rate of integration and ?-entropy of a class of analytic functions

Summary. We consider the problem of optimal quadratures for integrands f: [-1,1]--,~ which have an analytic extension f to an open disk D, of radius r about the origin such that Ill_-< 1 on D,. If r= 1, we show that the penalty for sampling the integrand at zeros of the Legendre polynomial of degree n rather than at optimal points, tends to infinity with n. In particular there is an "infinite" penalty for using Gauss quadrature. On the other hand, if r>l, Gauss quadrature is almost optimal. These results hold for both the worst-case and asymptotic settings.

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“…where N2G~ is given by (5). We prove that Gauss information is also very poor in the asymptotic setting.…”

Section: E(co*f) < E([n/4j)mentioning

confidence: 95%

“…(1) and the infimum in (1) is attained for a linear algorithm (quadrature). Bojanov [4,5] proved that e(n):= inf e(N,) > exp(-5 xl//n~) (…”

Section: Introductionmentioning

confidence: 99%

Is Gauss quadrature optimal for analytic functions?

1985

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“…The next natural step is to minimize Ep(x) over the set f2 (v The behaviour of Ep, N as N tends to infinity has been intensively studied during the last decade (see [1,3,[5][6][7][8]11]), after it was shown in [3] that E~, N is of order exp (-c l/N). This result was striking since previously best known was E2,N=O (N) in I-5].…”

Section: Introductionmentioning

confidence: 99%

A note on the optimal quadrature inH p

Andersson

Bojanov

1984

Numer. Math.

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“…In the context of the optimal quadrature formula the Blaschke products have been used by Osipenko [O95] and Bojanov [Bo73,Bo74] for the analytic functions on the unit circle. Therefore, it is very natural that the function for which the supremum in Lemma 8 is attained is, up to a conformal mapping f D , a finite Blaschke product.…”

Section: Remark 9 It Is Obvious Thatmentioning

confidence: 99%