Concerning the numerical methods presented in Section 7 of [1] for generating ray sequences of ra... more Concerning the numerical methods presented in Section 7 of [1] for generating ray sequences of rational functions rmn that are asymptotically optimal for the Zolotarev problem Zmn Zmn(E1, E2) := inf (supz~E' Ir(z)]) m =-+)% the discussion of Leja-Bagby points in part (b) requires clarification. Indeed, as pointed out by V. Totik, the argument given on pages 259-261 of [1] does not, in general, cover the hypothetical case when the sequence of signed measures {an } has more than one limit measure. That is, we have actually established in [1] the following result.
Proceedings of the American Mathematical Society, 2007
For certain sets in ${\rm {\bf R}}^d$ of integer Hausdorff dimension, we show that the limiting b... more For certain sets in ${\rm {\bf R}}^d$ of integer Hausdorff dimension, we show that the limiting behavior of the best-packing distance as well as the minimal $s$-energy for large $s$ is different for different subsequences of the cardinalities of the configurations.
For a continuous 2n-periodic real-valued function f, we investigate the asymptotic behavior of th... more For a continuous 2n-periodic real-valued function f, we investigate the asymptotic behavior of the zeros of the error f({})-s.({}), where s.({}) is the nth Fourier section. We prove that there is a subsequence {nk} for which such zeros (interpolation points) are uniformly distributed on [-n, n]. This extends previous results of gaff and Shekhtman. Moreover, results dealing with the maximal distance between consecutive zeros of f-s.. are obtained. The technique of proof involves coefficient estimates for lacunary trigonometric polynomials in terms of its Lq-norm on a subinterval.
Efficient curves" in the sense of the best rate of multivariate polynomial approximation to contr... more Efficient curves" in the sense of the best rate of multivariate polynomial approximation to contractive functions on these curves, were first introduced by D.J. Newman and L. Raymon in 1969. They proved that algebraic curves are efficient, but claimed that the exponential curve γ := {(t, e t): 0 t 1} is not. We prove to the contrary that this exponential curve and its generalization to higher dimensions are indeed efficient. We also investigate helical curves in R d and show that they too are efficient. Transcendental curves of the form {(t, t λ): δ t 1} are shown to be efficient for δ > 0, contradicting another claim of Newman and Raymon.
Heldermann Verlag where Bn is the collection of all Blaschke products of degree n. Denote by Bn E... more Heldermann Verlag where Bn is the collection of all Blaschke products of degree n. Denote by Bn E Bn a Blaschke product that attains the value ~n. We investigate the asymptotic behavior, as n-t 00, of the minimal Blaschke products Bn in the case when the measure p. with support E = [a, b) satisfies the Szegc:> condition: (b log(dp.{dx) dx >-00. la ";(x-a)(b-x) At the same time, we shall obtain results related to the convergence of best £1 approximants on the unit circle to the Markov function f(z)=~ {2 1r1 1 E Z-x by meromorphic functions of the form P {Q, where P belongs to the Hardy space HI of the unit disk and Q is a polynomial of degree at most n. We also include in an appendix a detailed treatment of a factorization theorem for Hardy spaces of the slit disk, which may be of independent interest.
Potential Theoretic Tools in Polynomial and Rational Approximation
Lecture Notes in Control and Information Science, 2006
Logarithmic potential theory is an elegant blend of real and complex analysis that has had a prof... more Logarithmic potential theory is an elegant blend of real and complex analysis that has had a profound effect on many recent developments in approximation theory. Since logarithmic potentials have a direct connection with polynomial and rational functions, the tools provided by classical potential theory and its extensions to cases when an external field (or weight) is present, have resolved some
We give a short and elementary proof of an inverse Bernstein-type inequality found by S. Khrushch... more We give a short and elementary proof of an inverse Bernstein-type inequality found by S. Khrushchev for the derivative of a polynomial having all its zeros on the unit circle. The inequality is used to show that equally-spaced points solve a min–max–min problem for the logarithmic potential of such polynomials. Using techniques recently developed for polarization (Chebyshev-type) problems, we show that this optimality also holds for a large class of potentials, including the Riesz potentials$1/r^{s}$with$s>0.$
Given a condenser (E, F) in the complex plane, let C(E, F) denote its capacity and let JL* = JLE ... more Given a condenser (E, F) in the complex plane, let C(E, F) denote its capacity and let JL* = JLE -JLj:. be the (signed) equilibrium distribution for (E, F) . Given a finite positive measure JL on EUF, let G(JLE) = exp (flog(dJL/dJLE)dp.E) and G(JLf.) = exp (f log(dp./dJLj:.) dp.j:.) .We show that for 0 < p, q < 00 and for an)1 rational function rn of order n (*) IlrnIILp(d/l,E)III/rnIlLq(d/l,F)?: e-n/C(E,F)GI/P(JLE)GI/q(JLf.) , which extends a classical result due toA. A. Gonchar. For a symmetric condenser we also obtain a sharp lower bound for IIrn -.J.IILp(d/l , EUF) ,where .J. = .J.(z) is equal to 0 on E and I on F. The question of exactness of (* ) and the relation to certain n-widths are also discussed.. ~ibliography: 16 titles.
Transactions of the American Mathematical Society, 2008
Given a compact d-rectifiable set A embedded in Euclidean space and a distribution ρ(x) with resp... more Given a compact d-rectifiable set A embedded in Euclidean space and a distribution ρ(x) with respect to d-dimensional Hausdorff measure on A, we address the following question: how can one generate optimal configurations of N points on A that are "well-separated" and have asymptotic distribution ρ(x) as N → ∞? For this purpose we investigate minimal weighted Riesz energy points, that is, points interacting via the weighted power law potential V = w(x, y) |x − y| −s , where s > 0 is a fixed parameter and w is suitably chosen. In the unweighted case (w ≡ 1) such points for N fixed tend to the solution of the best-packing problem on A as the parameter s → ∞.
We investigate the asymptotic behaviour, as N grows, of the largest minimal weighted pairwise dis... more We investigate the asymptotic behaviour, as N grows, of the largest minimal weighted pairwise distance between N points restricted to a rectifiable compact set embedded in Euclidean space, and we find the limit distribution of asymptotically optimal configurations. Bibliography: 23 titles.
Russian Academy of Sciences. Sbornik Mathematics, 1993
Given a finite positive measure Ji. on EUF,let G(Ji.E)=exp(Jlog(dJi./dJi.i-)dJi.i-) and G(Ji.F) =... more Given a finite positive measure Ji. on EUF,let G(Ji.E)=exp(Jlog(dJi./dJi.i-)dJi.i-) and G(Ji.F) = exp (Jlog(dJi./dJi.;')dJi.;') .We show that for 0 < p, q < 00 and for any rational function rn of order n (*) IlrnIILp(dp,E)lll/rnIILq(dp,F)?: e-n/C(E,F)GI/P(Ji.E)GI/q(Ji.F) , which extends a classical result due to A. A. Gonchar. For a symmetric condenser we also obtain a sharp lower bound for Ilrn-)..IILp(dp,EUF) , where).. =)..(z) is equal to 0 on E and 1 on F. The question of exactness of (*) and the relation to certain n-widths are also discussed. Bibliography: 16 titles.
Proceedings of the American Mathematical Society, 1989
In contrast to the behavior of best uniform polynomial approximants on [0, I] we show that if f E... more In contrast to the behavior of best uniform polynomial approximants on [0, I] we show that if f E C[O, I] there exists a sequence of polynomials {Pn} of respective degree ~ n which converges uniformly to f on [0, I] and geometrically fast at each point of [0, I] where f is analytic. Moreover we describe the best possible rates of convergence at all regular points for such a sequence.
For a bounded Jordan domain G with quasiconformal boundary L, two-sided estimates are obtained fo... more For a bounded Jordan domain G with quasiconformal boundary L, two-sided estimates are obtained for the error in best L 2 (G) polynomial approximation to functions of the form (z − τ) β , β > −1, and (z − τ) m log l (z − τ), m > −1, l / = 0, where τ ∈ L. Furthermore, Andrievskii's lemma that provides an upper bound for the L ∞ (G) norm of a polynomial p n in terms of the L 2 (G) norm of p n is extended to the case when a finite linear combination (independent of n) of functions of the above form is added to p n. For the case when the boundary of G is piecewise analytic without cusps, the results are used to analyze the improvement in rate of convergence achieved by using augmented, rather than classical, Bieberbach polynomial approximants of the Riemann mapping function of G onto a disk. Finally, numerical results are presented that illustrate the theoretical results obtained.
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Papers by Edward Saff