René Grothmann

On the zeros of sequences of polynomials

Abstract In this paper we generalize a result of Blatt, Saff, and Simkani on the limit distribution of zeros of sequences of polynomials. In a typical application these polynomials converge on a compact subset E of the complex plane. The highest coefficient of the polynomials plays an important role in the theorem of Blatt, Saff, and Simkani. In this paper we replace the behavior of the highest coefficient by the behavior of the sequence on some compact set in C \ E . Furthermore we show how this generalization can be applied to sequences of maximally convergent polynomials.

Erdös-Turán theorems on a system of Jordan curves and arcs

Erdös and Turán established in [4] a qualitative result on the distribution of the zeros of a monic polynomial, the norm of which is known on [−1, 1]. We extend this result to a polynomial bounded on a systemE of Jordan curves and arcs. If all zeros of the polynomial are real, the estimates are independent of the number of components ofE for any regular compact subsetE ofR. As applications, estimates for the distribution of the zeros of the polynomials of best uniform approximation and for the extremal points of the optimal error curve (generalizations of Kadecs theorem) are given.

On the Behavior of Zeros and Poles of Best Uniform Polynomial and Rational Approximants

We investigate the behavior of zeros of best uniform polynomial approximants to a function f, which is continuous in a compact set E ⊂ ℂ and analytic on E, but not on E. Our results are related to a recent theorem of Blatt, Saff, and Simkani which roughly states that the zeros of a subsequence of best polynomial approximants distribute like the equilibrium measure for E. In contrast, we show that there might be another subsequence with zeros essentially all tending to ∞. Also, we investigate near best approximants. For rational best approximants we prove that its zeros and poles cannot all stay outside a neighborhood of E, unless f is analytic on E.

Distribution of Interpolation Points

We show that interpolation to a function, analytic on a compact setE in the complex plane, can yield maximal convergence only if a subsequence of the interpolation points converges to the equilibrium distribution onE in the weak sense. Furthermore, we will derive a converse theorem for the case when the measure associated with the interpolation points converges to a measure onE, which may be different from the equilibrium measure.

Poles and Alternation Points in Real Rational Chebyshev Approximation

The distribution of equi-oscillation points (alternation points) for the error in best Chebyshev approximation on [−1,1] by rational functions is investigated. In general, the alternation points need not be dense in [−1,1] when rational functions of degree (n, m) are considered and asymptotically n/m → κ with κ ≥ 1. We show that the asymptotic behavior of the alternation points is closely related to the behavior of the poles of the rational approximants. Hence, poles of the rational approximations are attracting points of alternations such that the well-known equi-distribution for the polynomial case can be heavily disturbed.

On the real CF -method for polynomial approximation and strong unicity constants

Abstract We consider uniform polynomial approximation on [ −1, 1]. For the class of functions which are analytic in an ellipse with foci ± 1 and sum of semiaxes greater than 8.1722…, we prove several asymptotic results on the best approximation. We describe the CF -approximation method and prove that, for our class of functions, the CF -approximation is “not far away” from the best one. With the help of this result we show a Kadec type result on the alternants and prove a conjecture of Poreda on the strong uniqueness constants. Also we prove a lemma on the distance between the best approximation and a “good” approximating polynomial.

Detection of singularities using segment approximation

We discuss best segment approximation (with free knots) by poly- nomials to piecewise analytic functions on a real interval. It is shown that, if the degree of the polynomials tends to infinity and the number of knots is the same as the number of singularities of the function, then the optimal knots converge geometrically fast to the singularities. When the degree is held fixed and the number of knots tends to infinity, we study the asymptotic distribution of the optimal knots.

On Sign Changes in Weighted Polynomial \(L^1 \)-Approximation

AbstractFor

Growth Behavior and Zero Distribution of Maximally Convergent Rational Approximants

f \in L_{^w }^1 \left[ { - 1{\text{,}}1} \right]