On the Lebesgue constant of Leja sequences for the unit disk and its applications to multivariate interpolation (original) (raw)
Related papers
On the Lebesgue constant of Leja sequences for the complex unit disk and of their real projection
Journal of Approximation Theory, 2013
We consider Leja sequences of points for polynomial interpolation on the complex unit disk U and the corresponding sequences for polynomial interpolation on the real interval [−1, 1] obtained by projection. It was proved by Calvi and Phung in Calvi and Phung (2011, 2012) [3,4] that the Lebesgue constants for such sequences are asymptotically bounded in O(k log k) and O(k 3 log k) respectively, where k is the number of points. In this paper, we establish the sharper bound 5k 2 log k in the real interval case. We also give sharper bounds in the complex unit disk case, in particular 2k. Our motivation for producing such sharper bounds is the use of these sequences in the framework of adaptive sparse polynomial interpolation in high dimension. c
Interpolation and the Laguerre-Pólya class
A long standing open problem, known as the Karlin-Laguerre problem, in the study of the distribution of real zeros of a polynomial is to characterize all real sequences T={γ_k}_{k=0}^∞ such that they satisfy the property Z_c (T[p(x)])≤Z_c (p(x)), where Z_c(p(x)) denotes the number of non-real zeros of the real polynomial p(x)=∑_{k=0}^{n} a_k x_k and T(p(x))=∑_{k=0}^{n}γ_k a_k x_k. The main result of this paper shows that under a mild growth restriction, an entire function of exponential type f(z) for which the sequence T={f(k)}_{k=0}^{∞} satisfies the above condition must have only real zeros. The paper concludes with some applications to the Riemann hypothesis.
Lecture Notes in Computer Science, 2015
In the papers [6, 7] we have established linear and quadratic bounds, in k, on the growth of the Lebesgue constants associated with the k-sections of Leja sequences on the unit disc U and ℜ-Leja sequences obtained from the latter by projection into [−1, 1]. In this paper, we improve these bounds and derive sub-linear and sub-quadratic bounds. The main novelty is the introduction of a "quadratic" Lebesgue function for Leja sequences on U which exploits perfectly the binary structure of such sequences and can be sharply bounded. This yields new bounds on the Lebesgue constants of such sequences, that are almost of order √ k when k has a sparse binary expansion. It also yields an improvement on the Lebesgue constants associated with ℜ-Leja sequences.
On the Lebesgue constant for the Xu interpolation formula
Journal of Approximation Theory, 2006
In the paper , the author introduced a set of Chebyshev-like points for polynomial interpolation (by a certain subspace of polynomials) in the square [−1, 1] 2 , and derived a compact form of the corresponding Lagrange interpolation formula. In [1] we gave an efficient implementation of the Xu interpolation formula and we studied numerically its Lebesgue constant, giving evidence that it grows like O((log n) 2 ), n being the degree. The aim of the present paper is to provide an analytic proof that indeed the Lebesgue constant does have this order of growth.
On polynomial “interpolation” in L1
Journal of Approximation Theory, 1991
We study operators F from L, [ --K, x] into the space of trigonometric polynomials of degree m > n that satisfy II additional conditions OodFf) = g,(f), for all FE FH and all choices of points t,, . . . . t,,. 24
Order of uniform approximation by polynomial interpolation in the complex plane and beyond
Indagationes Mathematicae, 2023
Let X denote a compact set in the complex plane C and z n ∶= {z n,j } * * n ∶= {z * * n,j } n j=0 of the Fejér points on γ will be described to assure the growth rate of L z * * n to be exactly O(log 2 (n)). Dedicated to Prof. Jaap Korevaar on the occasion of his 100-th birthday! Keywords Lebesgue constants; Marcinkiewicz-Zygmund inequalities; Fejér and Fekete points; Polynomial interpolation and approximation with restricted zeros; Distribution of elections and total energy; Super-resolution pointmass recovery.
On P ´ Al-Type Interpolation II
In this paper, we study the convergence of Pál-type interpolation on two sets of non-uniformly distributed zeros on the unit circle, which are obtained by projecting vertically the nodes of the real line.