Asymptotics for the zeros of the partial sums of $\bf e^z$. I.

Carpenter, Amos J.; Varga, Richard S.; Waldvogel, Jörg

doi:10.1216/rmjm/1181072998

Cited by 24 publications

(39 citation statements)

References 7 publications

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“…It is possible to obtain more precise asymptotics for e −nz s n (nz) than that of (2.4), as is done, for example in Carpenter et al [3] and [4], but we do not pursue this direction here.…”

Section: Igor E Pritsker and Richard S Vargamentioning

confidence: 99%

The Szego curve, zero distribution and weighted approximation

Pritsker

Varga

1997

Trans. Amer. Math. Soc.

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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.

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Section: Igor E Pritsker and Richard S Vargamentioning

confidence: 99%

The Szego curve, zero distribution and weighted approximation

Pritsker

Varga

1997

Trans. Amer. Math. Soc.

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“…The theorem has numerous applications, which range from asymptotics of partial sums of power series [11] or a local-global stability principle for discrete-time systems [28], to coding theory [13], the economic theory of 10 depreciation and reinvestment [41], stability analysis of delay filters [36] and to models of high energy collisions [10] in physics. In this paper we are concerned with an extension of the Eneström-Kakeya theorem to operators in Hilbert space, which is different from the ones given by Furuta and Nakamura [21] and by Fuji and Kubo [18].…”

Section: 2)mentioning

confidence: 99%

Spectraloid operator polynomials, the approximate numerical range and an Eneström–Kakeya theorem in Hilbert space

Swoboda

Wimmer

2010

Studia Math.

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Mathematical Subject Classifications (2000): 47A10, 47A56, 47A12Keywords: Operator polynomials, Eneström-Kakeya theorem, normal approximate eigenvalues, semisimple eigenvalues, Jordan chains, approximate point spectrum, residual spectrum, spectral radius, numerical range, numerical radius, spectraloid operator. Running title: Operator polynomialsAbstract: In this paper we study a class of operator polynomials in Hilbert space, which are spectraloid in the sense that spectral radius and numerical radius coincide. The focus is on the spectrum in the boundary of the numerical range. As an application the Eneström-Kakeya-Hurwitz theorem on zeros of real polynomials is generalized to Hilbert space.

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“…This problem was posed and solved for f (z) = e z in the classical paper of Szegö [18]. Szegö proved that as n → ∞, the rescaled zeros, 4) approach the curve, Γ = {ζ : |e 1−ζ ζ| = 1, |ζ| ≤ 1}, (1.5) on the complex plane, and the limiting distribution of the zeros on Γ is the measure of the maximal entropy, the preimage of the uniform measure on the circle under the Riemann map. Precise asymptotics of the zeros of sections of e z and the sections themselves were obtained in the works of Buckholtz [3], Newman and Rivlin [13], Carpenter, Varga, and Waldvogel Date: August 28, 2018.…”

Section: Introductionmentioning

confidence: 99%

“…1 [4], Pritsker and Varga [15]. The absence of zeros in some parabolic domains on the complex plane was established in the works of Newman and Rivlin [13] and Saff and Varga [16].…”

Section: Introductionmentioning

confidence: 99%

Zeros of sections of exponential sums

Bleher

Mallison

2006

International Mathematics Research Notices

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Abstract. We derive the large n asymptotics of zeros of sections of a generic exponential sum. We divide all the zeros of the n-th section of the exponential sum into "genuine zeros", which approach, as n → ∞, the zeros of the exponential sum, and "spurious zeros", which go to infinity as n → ∞. We show that the spurious zeros, after scaling down by the factor of n, approach a "rosette", a finite collection of curves on the complex plane, resembling the rosette. We derive also the large n asymptotics of the "transitional zeros", the intermediate zeros between genuine and spurious ones. Our results give an extension to the classical results of Szegö about the large n asymptotics of zeros of sections of the exponential, sine, and cosine functions.

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Asymptotics for the zeros of the partial sums of $\bf e^z$. I.

Cited by 24 publications

References 7 publications

The Szego curve, zero distribution and weighted approximation

The Szego curve, zero distribution and weighted approximation

Spectraloid operator polynomials, the approximate numerical range and an Eneström–Kakeya theorem in Hilbert space

Zeros of sections of exponential sums

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