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Asymptotics of Polynomials Orthogonal on a Cross with a Jacobi-Type Weight

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Abstract

We investigate asymptotic behavior of polynomials \( Q_n(z) \) satisfying non-Hermitian orthogonality relations

$$\begin{aligned} \int _\Delta s^kQ_n(s)\rho (s){\mathrm {d}}s =0, \quad k\in \{0,\ldots ,n-1\}, \end{aligned}$$

where \( \Delta := [-\,a,a]\cup [-\,{\mathrm {i}}b,{\mathrm {i}}b] \), \( a,b>0 \), and \( \rho (s) \) is a Jacobi-type weight.

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Notes

  1. 1.

    Here and in what follows we state jump relations understanding that they hold outside the points of self-intersection of the considered arcs.

  2. 2.

    \( g_\Delta (z;\infty ) \) is equal to zero on \( \Delta \), is positive and harmonic in \( {\mathbb {C}}{\setminus }\Delta \), and satisfies \( g(z;\infty )=\log |z|+{\mathcal {O}}(1) \) as \(z\rightarrow \infty \).

  3. 3.

    In what follows we write \( |g_1(z)|\sim |g_2(z)| \) as \( z\rightarrow z_0 \) if there exists a constant \( C>1 \) such that \( C^{-1}|g_1(z)|\le |g_2(z)| \le C|g_1(z)| \) for all \( z \) close to \( z_0 \).

  4. 4.

    Hereafter, we set \(\sigma _3 := \left( \begin{matrix} 1 &{} 0 \\ 0 &{} -1 \end{matrix}\right) \) and \( {\varvec{I}} \) to be the identity matrix.

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Correspondence to Maxim L. Yattselev.

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The research was supported in part by a Grant from the Simons Foundation, CGM-354538.

Communicated by Laurent Baratchart.

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Barhoumi, A., Yattselev, M.L. Asymptotics of Polynomials Orthogonal on a Cross with a Jacobi-Type Weight. Complex Anal. Oper. Theory 14, 9 (2020) doi:10.1007/s11785-019-00962-7

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Keywords

  • Non-Hermitian orthogonality
  • Strong asymptotics
  • Padé approximation
  • Riemann–Hilbert analysis

Mathematics Subject Classification

  • 42C05
  • 41A20
  • 41A21