Polynomial approximation on the boundary and strictly inside

We investigate the possibility of approximating a function on a compact setK of the complex plane in such a way that the rate of approximation is almost optimal onK, and the rate inside the interior ofK is faster than on the whole ofK. We show that ifK has an external angle smaller than π at some point z_o∈δK, then geometric convergence insideK is possible only for functions that are analytic at z_o. We also consider the possibility of approximation rates of the form exp(?cn ^β), for approximation insideK, where β is related to the largest external angle ofK. It is also shown that no matter how slowly the sequence {γ _n } tends to zero, there is aK and a Lip β, β<1, functionf such that approximation insideK cannot have order {γ _n }.     

Extensions of Szegö's theory of orthogonal polynomials,III

Let {ø _n (dμ)} be a system of orthonormal polynomials on the unit circle with respect to a measuredμ. Szegö's theory is concerned with the asymptotic behavior ofø _n (dμ) when logμ′∈L ¹. In what follows we will discuss the asymptotic behavior of the ratioø _n (dμ ₂)/ø _n (dμ ₁) on the unit circle whendμ ₁ anddμ ₂ are close in a sense (e.g.,dμ ₂=gdμ ₁, where g≥0 is such thatQ(e ^it )g(t) andQ(e ^it )/g(t) are bounded for a suitable polynomialQ) and μ ₁ ^′ >0 almost everywhere or (a somewhat weaker requirement) lim _n→∞Φ _n (dμ ₁,0)=0 for the monic polynomial Φ _n . The asymptotic behavior of the same fraction outside the unit circle was discussed in an earlier paper.     

Sharp Nikolskii inequalities with exponential weights

w _a(x)=exp(–x^a), xR, a0. , N _n (a,p,q) — (2), _n P _nw_ap, CN_n(a,p, q)P_nw_aq. , — , {P _n}, .

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This material is based upon research supported by the National Science Foundation under Grant No. DMS-84-19525, by the United States Information Agency under Senior Research Fulbright Grant No. 85-41612, and by the Hungarian Ministry of Education (first author). The work was started while the second author visited The Ohio State University between 1983 and 1985, and it was completed during the first author's visit to Hungary in 1985.     

Weighted polynomial inequalities

For the weights exp (?|x|^λ), 0<λ≤1, we prove the exact analogue of the Markov-Bernstein inequality. The Markov-Bernstein constant turns out to be of order logn for λ=1 and of order 1 for 0<λ<1. The proof is based on the solution of the problem of how fast a polynomialP _n can decrease on [?1,1] ifP _n (0)=1. The answer to this problem has several other consequences in different directions; among others, it leads to a general theorem about the incompleteness of the set of polynomials in weightedL ^p spaces.     

Saturation of the Shepard operators

We discuss the Shepard operators S_n (f; x) in this paper and establish the saturation of the sequence {S_n f} _n-1 ^∞ , as well as investigate some related questions. The research of this author was supported in part by the Hungarian Science Foundation for Research, Grant. N ^o . 1157.     

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.     

A conjecture of Gy. Petruska

Summary We give a negative answer to a conjecture of Gy. Petruska.     

The norm of minimal polynomials on several intervals

Using works of Franz Peherstorfer, we examine how close the $n$th Chebyshev number for a set $E$ of finitely many intervals can get to the theoretical lower limit $2\mathrm{cap}{\left(E\right)}^n$.     

Sharp converse theorem ofL p -polynomial approximation

For algebraic polynomial approximation on [?1,1] the analogue of the Zygmund-Timan type converse Marchaud inequality is proved. These are the exact converse estimates inL ^p spaces when 1<p<∞. The proof forp>2 uses Hirschman's multiplier theory, while forp相似文献