The Szego curve, zero distribution and weighted approximation

Abstract

Abstract. In 1924, Szegő showed that the zeros of the normalized partial sums, sn(nz), of e z tended to what is now called the Szegő curve S, where S := z ∈ C : |ze 1−z | = 1 and |z| ≤ 1 .Using modern methods of weighted potential theory, these zero distribution results of Szegő can be essentially recovered, along with an asymptotic formula for the weighted partial sums {e −nz sn(nz)} ∞ n=0 . We show that G := Int S is the largest universal domain such that the weighted polynomials e −nz Pn(z) are dense in the set of functions analytic in G. As an example of such results, it is shown that if f (z) is analytic in G and continuous on G with f (1) = 0, then there is a sequence of polynomials {Pn(z)} ∞ n=0 , with deg Pn ≤ n, such that limwhere · G denotes the supremum norm on G. Similar results are also derived for disks.

Zero Attractors of Partition Polynomials

“…Since the asymptotics for F n (z) depends primarily on the heaviest singularity, a Stokes phenomenon [22] appears forcing the roots fall near the Stokes/anti-Stokes lines. This is analogous to the phenomenon observed by the well studied Szego curve [28] and the Appell polynomials [6,14].…”

Erlang Redux: An Ansatz Method for Solving the M/M/m Queue

“…Conversely, zeros of the corrected denominator in (9) lie on the interior of the unit disk. As m increases, those zeros move further away from the circumference and converge on the Szegő bound [4,15]. Even without understanding the mathematical construction, Fig.…”

The Szegő curve and Laguerre polynomials with large negative parameters

“…To this end, consider the function φ(z) = ze 1−z . It is easy to see that φ conformally maps G r onto the disk D r = {w ∈ C/|w| < r} , 0 ≤ r < ∞ , in the w-plane (see [20] and [15]). Thus, from (2.2), we have:…”

“…which is a closed curve around the origin passing through z = 1 and crossing once the negative real semiaxis (−∞, 0) (see Figure 1). See also [15] for a detailed study of the Szegö curve and some related problems in approximation of functions. Recently, T. Kriecherbauer et al [6] obtained uniform asymptotic expansions for the partial sums of the exponential series by means of the Riemann-Hilbert analysis.…”