Generalized Christoffel functions for power orthogonal polynomials

In this paper we extend the Christoffel functions to the case of power orthogonal polynomials. The existence and uniqueness as well as some properties are given.     

Christoffel functions and mean convergence for Lagrange interpolation for exponential weights

For exponential weights, a necessary condition of weighted mean convergence for Lagrange interpolation is given.     

Generalized Christoffel functions and error of positive quadrature

The author gives some upper and lower bounds for the generalized Christoffel functions related to a Ditzian-Totik generalized weight. As an application, an error estimate of Gauss quadrature formula inL ₁-weighted norm is derived.Dedicated to Prof. Luigi Gatteschi on the occasion of his 70th birthdayWork sponsored by MURST 40%.     

Christoffel functions,orthogonal polynomials,and Nevai's conjecture for Freud weights

We obtain upper and lower bounds for Christoffel functions for Freud weights by relatively new methods, including a new way to estimate discretization of potentials. We then deduce bounds for orthogonal polynomials on thereby largely resolving a 1976 conjecture of P. Nevai. For example, let W:=e ^–Q, whereQ: is even and continuous in, Q^" is continuous in (0, ) andQ ^'>0 in (0, ), while, for someA, B,

Erratum: Christoffel functions,orthogonal polynomials,and Nevai's conjecture for Freud weights

Communicated by Edward B. Saff     

On generalized Christoffel functions

The generalized Christoffel function λ _p,q,n (dμ;x) (0<p<∞, 0≦q<∞) with respect to a measure dμ on R is defined by

$\lambda_{p,q,n}(d\mu;x)=\inf_{Q\in\mathbf{P}_{n-1},\ Q(x)=1}\int_{\mathbf{R}} \big|Q(t)\big|^p {|t-x|}^q\, d\mu(t).$

The novelty of our definition is that it contains the factor |t?x| ^q , which is of particular interest. Its properties are discussed and estimates are given. In particular, upper and lower bounds for generalized Christoffel functions with respect to generalized Jacobi weights are also provided.

     

Asymptotics for Christoffel functions of planar measures

We prove a version of asymptotics of Christoffel functions with varying weights for a general class of sets E in, and measures μ on the complex plane ℂ. This class includes all regular measures μ in the sense of Stahl-Totik [18] on regular compact sets E in ℂ and even allows varying weights. Our main theorems cover some known results for subsets of the real line ℝ. In particular, we recover information in the case of E = ℝ with Lebesgue measure dx and weight w(x) = exp(−Q(x)) where Q(x) is a nonnegative even degree polynomial having positive leading coefficient. Supported in part by an NSERC of Canada grant     

Asymptotics for Christoffel functions for general measures on the real line

We consider asymptotics of Christoffel functions for measures ν with compact support on the real line. It is shown that under some natural conditionsn times thenth Christoffel function has a limit asn→∞ almost everywhere on the support, and the limit is the Radon-Nikodym derivative of ν with respect to the equilibrium measure of the support of ν. The case in which the support is an interval was settled previously by A. Máté, P. Nevai and the author. The present paper solves the general problem. Work was supported by the National Science Foundation, DMS 9801435 and by the Hungarian National Science Foundation for Research, T/022983.     

How poles of orthogonal rational functions affect their Christoffel functions

We show that even a relatively small number of poles of a sequence of orthogonal rational functions approaching the interval of orthogonality, can prevent their Christoffel functions from having the expected asymptotics. We also establish a sufficient condition on the rate for such asymptotics, provided the rate of approach of the poles is sufficiently slow. This provides a supplement to recent results of the authors where poles were assumed to stay away from the interval of orthogonality.     

Christoffel functions for doubling measures on quasismooth curves and arcs

Matching two-sided estimates are given for Christoffel functions associated with a doubling measure ν over a quasismooth curve or arc. The size of the the n-th Christoffel function at a point z is given by the ν-measure of the largest disk about z which lies within the 1/n-level line of the Green’s function. The main theorem contains as special case all previously known weak asymptotics for Christoffel functions, and it also gives their size in explicit form about smooth corners. Applications are given for estimating the size of orthonormal polynomials and for Nikolskii-type inequalities.     

Asymptotics of Christoffel functions on arcs and curves

For a system of smooth Jordan curves and arcs asymptotics for Christoffel functions is established. A separate new method is developed to handle the upper and lower estimates. In the course to the upper bound a theorem of Widom on the norm of Chebyshev polynomials is generalized.     

Christoffel-type functions for -orthogonal polynomials for Freud weights

This paper gives upper and lower bounds of the Christoffel-type functions , for the m-orthogonal polynomials for a Freud weight W=e^-Q, which are given as follows. Let a_n=a_n(Q) be the nth Mhaskar–Rahmanov–Saff number, φ_n(x)=max{n^-2/3,1-|x|/a_n}, and d>0. Assume that QC(R) is even, , and for some A,B>1

Then for xR

and for |x|a_n(1+dn^-2/3)

     

Christoffel functions and universality on the boundary of the ball

We establish asymptotics for Christoffel functions, and universality limits, associated with multivariate orthogonal polynomials, on the boundary of the unit ball in ? ^d .     

Generalized hamming weights and equivalences of codes

It is proved that any linear isomorphism between two linear codes which preserves a generalizedHamming weight is a monomial equivalence.which is an extension of a theorem of MacWilliams.     

On the order of reflection coefficients for Generalized Jacobi weights

This paper generalizes some results of L. B. Golinskii [4] on the asymptotic behaviour of reflection coefficients associated with generalized Jacobi weight functions.     

Generalized Lebesgue points for Sobolev functions

In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point x in a metric measure space (X, d, μ) is called a generalized Lebesgue point of a measurable function f if the medians of f over the balls B(x, r) converge to f(x) when r converges to 0. We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function. We show that a function f ∈ M ^s,p (X), 0 < s ≤ 1, 0 < p < 1, where X is a doubling metric measure space, has generalized Lebesgue points outside a set of \(\mathcal{H}^h\)-Hausdorff measure zero for a suitable gauge function h.     

Generalized Ahlfors functions

Let be a bordered Riemann surface with genus and boundary components. Let be a smooth family of smooth Jordan curves in which all contain the point 0 in their interior. Let and let be the family of all bounded holomorphic functions on such that and for almost every . Then there exists a smooth up to the boundary holomorphic function with at most zeros on so that for every and such that for every . If, in addition, all the curves are strictly convex, then is unique among all the functions from the family .