Zero Distribution Of Orthogonal Polynomials Research Articles

Zero distribution of complex orthogonal polynomials with respect to exponential weights

We study the limiting zero distribution of orthogonal polynomials with respect to some particular exponential weights e−nV(z) along contours in the complex plane. We are especially interested in the question under which circumstances the zeros of the orthogonal polynomials accumulate on a single analytic arc (one cut case), and in which cases they do not. In a family of cubic polynomial potentials V(z)=−iz33+iKz, we determine the precise values of K for which we have the one cut case. We also prove the one cut case for a monomial quintic V(z)=−iz55 on a contour that is symmetric in the imaginary axis.

A note on asymptotic zero distribution of orthogonal polynomials

We strengthen a theorem of Kuijlaars and Serra Capizzano on the distribution of zeros of a sequence of orthogonal polynomials { p n } n=1 ∞ for which the coefficients in the three term recurrence relation are clustered at finite points. The proof uses a matrix argument motivated by a theorem of Tyrtyshnikov.

Asymptotic Zero Distribution of Orthogonal Polynomials with Discontinuously Varying Recurrence Coefficients

The zero distribution of orthogonal polynomials pn, N,n=0, 1,… generated by recurrence coefficients an, N and bn, N depending on a parameter N has been recently considered by Kuijlaars and Van Assche under the assumption that an, N and bn, N behave like a(n/N) and b(n/N), respectively, where a(·) and b(·) are continuous functions. Here, we extend this result by allowing a(·) and b(·) to be measurable functions so that the presence of possible jumps is included. The novelty is also in the sense of the mathematical tools since, instead of applying complex analysis arguments, we use recently developed results from asymptotic matrix theory due to Tyrtyshnikov, Serra Capizzano, and Tilli.

The Asymptotic Zero Distribution of Orthogonal Polynomials with Varying Recurrence Coefficients

We study the zeros of orthogonal polynomials pn, N, n=0, 1, …, that are generated by recurrence coefficients an, N and bn, N depending on a parameter N. Assuming that the recurrence coefficients converge whenever n, N tend to infinity in such a way that the ratio n/N converges, we show that the polynomials pn, N have an asymptotic zero distribution as n/N tends to t>0 and we present an explicit formula for the limiting measure. This formula contains the asymptotic zero distri- butions for various special classes of orthogonal polynomials that were found earlier by different methods, such as Jacobi polynomials with varying parameters, discrete Chebyshev polynomials, Krawtchouk polynomials, and Tricomi–Carlitz polynomials. We also give new results on zero distributions of Charlier polynomials, Stieltjes–Wigert polynomials, and Lommel polynomials.

Asymptotic zero distribution of orthogonal polynomials in sinusoidal frequency estimation

It is shown that the zeros of polynomials orthogonal with respect to certain positive measures on the unit circle have an asymptotic distribution which is uniform on a circle of radius less than or equal to one. These results explain some phenomena observed when various linear prediction methods are used to estimate sinusoidal frequencies. They also describe the asymptotic zero distribution of the prediction error filter polynomials for a class of time series including autoregressive moving average (ARMA) models.