Lebesgue constant for Lagrange interpolation on equidistant nodes

Properties of Lebesgue function for Lagrange interpolation on equidistant nodes are investigated. It is proved that Lebesgue function can be formulated both in terms of a hypergeometric function 2F1 and Jacobt polynomials. Moreover, an integral expression of Lebesgue function is also obtained and the asymptotic behavior of Lebesgue constant is studied.     

The Lebesgue constant for Lagrange interpolation on equidistant nodes

Summary Forn=1, 2, 3, ..., let _n denote the Lebesgue constant for Lagrange interpolation based on the equidistant nodesx _{k, n} =k, k=0, 1, 2, ...,n. In this paper an asymptotic expansion for log _n is obtained, thereby improving a result of A. Schönhage.     

On the Lebesgue constant of barycentric rational interpolation at equidistant nodes

Recent results reveal that the family of barycentric rational interpolants introduced by Floater and Hormann is very well-suited for the approximation of functions as well as their derivatives, integrals and primitives. Especially in the case of equidistant interpolation nodes, these infinitely smooth interpolants offer a much better choice than their polynomial analogue. A natural and important question concerns the condition of this rational approximation method. In this paper we extend a recent study of the Lebesgue function and constant associated with Berrut’s rational interpolant at equidistant nodes to the family of Floater–Hormann interpolants, which includes the former as a special case.     

Lower bound in the Bernstein inequality for the first derivative of algebraic polynomials

We prove that max |p′(x)|, where p runs over the set of all algebraic polynomials of degree not higher than n ≥ 3 bounded in modulus by 1 on [−1, 1], is not lower than ( n - 1 ) \mathord

Lebesgue functions corresponding to a family of Lagrange interpolation polynomials

In this paper we obtain various explicit forms of the Lebesgue function corresponding to a family of Lagrange interpolation polynomials defined at an even number of nodes. We study these forms by using the derivatives up to the second order inclusive. We estimate exact values of Lebesgue constants for this family from below and above in terms of known parameters. In a particular case we obtain new convenient formulas for calculating these estimates.     

A complete description of the Lebesgue functions for classical lagrange interpolation polynomials

Explicit forms of Lebesgue functions are not described in the mathematical literature by now. This issue is related to the problem of reducing sums of modules of fundamental polynomials or the corresponding Dirichlet kernels. That is why the complete study of graphs of Lebesgue functions remains a complicated topical problem in the theory of approximation. In this paper we solve the mentioned problems both for odd and even numbers of interpolation nodes. We find explicit forms of Lebesgue functions and study them by means of the differential calculus. All mentioned forms are new.     

On the Lebesgue constant of Berrut’s rational interpolant at equidistant nodes

It is well known that polynomial interpolation at equidistant nodes can give bad approximation results and that rational interpolation is a promising alternative in this setting. In this paper we confirm this observation by proving that the Lebesgue constant of Berrut’s rational interpolant grows only logarithmically in the number of interpolation nodes. Moreover, the numerical results suggest that the Lebesgue constant behaves similarly for interpolation at Chebyshev as well as logarithmically distributed nodes.     

Asymptotic formulas for Lebesgue functions corresponding to the family of Lagrange interpolation polynomials

Mathematical Notes - The asymptotic behavior of Lebesgue functions of trigonometric Lagrange interpolation polynomials constructed on an even number of nodes is studied. For these functions,...     

Lower bound for the maximum of a stochastic process

The first author's participation in the preparation of this paper was supported by the Russian Foundation for Basic Research (Grant No. 93-01-00240).     

An improved lower bound for the expected number of real zeros of a random algebraic polynomials

In this note we estimate the lower bound of the average number of real zeros of a random algebraic polynomials when the random coefficients are standard normal random variables     

An interpolation process on the roots of the integrated Legendre polynomials

Пусть ?1=х _n,n <x _n?1,n <...<x _1,n =1 корн и многочлена $$\Pi _n \left( x \right) = - \left( {n - 1} \right)n\mathop \smallint \limits_{ - 1}^x P_{n - 1} \left( t \right)dt,$$ гдеP _n?1 — многочлен Леж андра степени (n?1) иx _i,n ^* (i=1, 2, ...,n ? 1) корни многочлен аΠ _n ′. В работе доказываетс я теорема о сходимост и многочленовR _n (n=2, 4, 6, ...), удо влетворяющих следующим условиям: гдеy _i,n иy _i,n ′ - заданные си стемы значений. Неулучшаемость теор емы также доказана.     

Degree-uniform lower bound on the weights of polynomials with given sign function

A Boolean function f: {0, 1} ⁿ → {0, 1} is called the sign function of an integer polynomial p of degree d in n variables if it is true that f(x) = 1 if and only if p(x) > 0. In this case the polynomial p is called a threshold gate of degree d for the function f. The weight of the threshold gate is the sum of the absolute values of the coefficients of p. For any n and d ≤ D ≤ $\frac{{\varepsilon n^{1/5} }} {{\log n}} $\frac{{\varepsilon n^{1/5} }} {{\log n}} we construct a function f such that there is a threshold gate of degree d for f, but any threshold gate for f of degree at most D has weight 2^{(dn)d /D4d} 2^{(\delta n)^d /D^{4d} } , where ɛ > 0 and δ > 0 are some constants. In particular, if D is constant, then any threshold gate of degree D for our function has weight 2^{W(nd )}2^{\Omega (n^d )} . Previously, functions with these properties have been known only for d = 1 (and arbitrary D) and for D = d. For constant d our functions are computable by polynomial size DNFs. The best previous lower bound on the weights of threshold gates for such functions was 2^Ω(n). Our results can also be translated to the case of functions f: {−1, 1} ⁿ → {−1, 1}.     

Properties of interpolatory quadrature with equidistant nodes on the unit circle

In this paper, we study the computation of the moments associated to rational weight functions given as a power spectrum with known or unknown poles of any order in the interior of the unit disc. A recursive algebraic procedure is derived that computes the moments in a finite number of steps. We also study the associated interpolatory quadrature formulas with equidistant nodes on the unit circle. Explicit expressions are given for the positive quadrature weights in the case of a polynomial weight function. For rational weight functions with simple poles, mostly real or uniformly distributed on a circle in the open unit disc, we also obtain expressions for the quadrature weights and sufficient conditions that guarantee that they are positive. The Poisson kernel is a simple example of a rational weight function, and in the last section, we derive an asymptotic expansion of the quadrature error.