Supercritical regime for the kissing polynomials

Celsus, Andrew F.; Silva, Guilherme L. F.

doi:10.1016/j.jat.2020.105408

Cited by 5 publications

(34 citation statements)

References 42 publications

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“…Moreover, the quadratic differential listed above differs from that of Ref. 25 by a factor of 4. For more details, we refer the reader to Sections 4 and 5 of Ref.…”

Section: Introductionmentioning

confidence: 78%

“…The analysis of the varying‐weight Kissing polynomials for

t > t_{c}

was undertaken in Ref. 25. Again, using the Riemann–Hilbert approach for these polynomials, the authors were able to show that there exist analytic arcs

γ_{m, 0} false(t false)

and

γ_{m, 1} false(t false)

such that the zeros of the varying‐weight Kissing polynomials accumulate on

γ_{m, 0} \cup γ_{m, 1}

n \to \infty

.…”

Section: Introductionmentioning

confidence: 99%

“…and

ℜ h (z) > 0

for

z

in close proximity on either side of

γ_{m, 0}

and

γ_{m, 1}

.Remark The

h

function described above is given by

h (z; s) = - 2 ϕ (z) + i κ

in the notation of, 25 where

κ \in R

is a real constant of integration. Moreover, the quadratic differential listed above differs from that of Ref.…”

Section: Introductionmentioning

confidence: 99%

“…We also point out that a further continuation of the work in Refs. 24 and 25 was carried out in Ref. 26, where varying‐weight Kissing polynomials with a Jacobi‐type weight were considered.…”

Section: Introductionmentioning

confidence: 99%

“…Another natural generalization of the works 24,25 is to allow

t

to take on complex values. That is, instead of considering the polynomials defined in () with

t \in {double-struckR}_{+} ∖ false{t_{c} false}

, we consider monic polynomials

\int_{- 1}^{1} p_{n} false(z; s false) z^{k} e^{- n f (z; s)} d z = 0, k = 0, 1, \dots, n - 1,

where

f (z; s) = s z

and

s \in C

is arbitrary, as introduced in ().…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Global‐phase portrait and large‐degree asymptotics for the Kissing polynomials

2021

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We study a family of monic orthogonal polynomials that are orthogonal with respect to the varying, complex‐valued weight function, exp(nsz), over the interval [−1,1], where s∈C is arbitrary. This family of polynomials originally appeared in the literature when the parameter was purely imaginary, that is, s∈iR, due to its connection with complex Gaussian quadrature rules for highly oscillatory integrals. The asymptotics for these polynomials as n→∞ have recently been studied for s∈iR, and our main goal is to extend these results to all s in the complex plane. We first use the technique of continuation in parameter space, developed in the context of the theory of integrable systems, to extend previous results on the so‐called modified external field from the imaginary axis to the complex plane minus a set of critical curves, called breaking curves. We then apply the powerful method of nonlinear steepest descent for oscillatory Riemann–Hilbert problems developed by Deift and Zhou in the 1990s to obtain asymptotics of the recurrence coefficients of these polynomials when the parameter s is away from the breaking curves. We then provide the analysis of the recurrence coefficients when the parameter s approaches a breaking curve, by considering double scaling limits as s approaches these points. We see a qualitative difference in the behavior of the recurrence coefficients, depending on whether or not we are approaching the points s=±2 or some other points on the breaking curve.

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“…Moreover, the quadratic differential listed above differs from that of Ref. 25 by a factor of 4. For more details, we refer the reader to Sections 4 and 5 of Ref.…”

Section: Introductionmentioning

confidence: 78%

“…The analysis of the varying‐weight Kissing polynomials for

t > t_{c}

was undertaken in Ref. 25. Again, using the Riemann–Hilbert approach for these polynomials, the authors were able to show that there exist analytic arcs

γ_{m, 0} false(t false)

and

γ_{m, 1} false(t false)

such that the zeros of the varying‐weight Kissing polynomials accumulate on

γ_{m, 0} \cup γ_{m, 1}

n \to \infty

.…”

Section: Introductionmentioning

confidence: 99%

“…and

ℜ h (z) > 0

for

z

in close proximity on either side of

γ_{m, 0}

and

γ_{m, 1}

.Remark The

h

function described above is given by

h (z; s) = - 2 ϕ (z) + i κ

in the notation of, 25 where

κ \in R

is a real constant of integration. Moreover, the quadratic differential listed above differs from that of Ref.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Another natural generalization of the works 24,25 is to allow

t

to take on complex values. That is, instead of considering the polynomials defined in () with

t \in {double-struckR}_{+} ∖ false{t_{c} false}

, we consider monic polynomials

\int_{- 1}^{1} p_{n} false(z; s false) z^{k} e^{- n f (z; s)} d z = 0, k = 0, 1, \dots, n - 1,

where

f (z; s) = s z

and

s \in C

is arbitrary, as introduced in ().…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Global‐phase portrait and large‐degree asymptotics for the Kissing polynomials

2021

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Strong asymptotics of Jacobi-type kissing polynomials

Barhoumi

2021

Integral Transforms and Special Functions

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We investigate asymptotic behavior of polynomials p ω n (z) satisfying varying non-Hermitian orthogonality relationsis holomorphic and non-vanishing in a certain neighborhood in the plane. These polynomials are an extension of so-called kissing polynomials (α = β = 0) introduced in [1] in connection with complex Gaussian quadrature rules with uniform good properties in ω. The analysis carried out here is an extension of what was done in [2,3], and depends heavily on those works.

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The kissing polynomials and their Hankel determinants

Celsus

Deaño

Huybrechs

et al. 2021

Transactions of Mathematics and Its Applications

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In this paper, we investigate algebraic, differential and asymptotic properties of polynomials $p_n(x)$ that are orthogonal with respect to the complex oscillatory weight $w(x)=\mathrm {e}^{\mathrm {i}\omega x}$ on the interval $[-1,1]$, where $\omega>0$. We also investigate related quantities such as Hankel determinants and recurrence coefficients. We prove existence of the polynomials $p_{2n}(x)$ for all values of $\omega \in \mathbb {R}$, as well as degeneracy of $p_{2n+1}(x)$ at certain values of $\omega $ (called kissing points). We obtain detailed asymptotic information as $\omega \to \infty $, using recent theory of multivariate highly oscillatory integrals, and we complete the analysis with the study of complex zeros of Hankel determinants, using the large $\omega $ asymptotics obtained before.

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Supercritical regime for the kissing polynomials

Cited by 5 publications

References 42 publications

Global‐phase portrait and large‐degree asymptotics for the Kissing polynomials

Global‐phase portrait and large‐degree asymptotics for the Kissing polynomials

Strong asymptotics of Jacobi-type kissing polynomials

The kissing polynomials and their Hankel determinants

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