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Extremal problems in de Branges spaces: the case of truncated and odd functions

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Abstract

In this paper we find extremal one-sided approximations of exponential type for a class of truncated and odd functions with a certain exponential subordination. These approximations optimize the \(L^1({\mathbb {R}}, |E(x)|^{-2}\mathrm{d}x)\)-error, where \(E\) is an arbitrary Hermite–Biehler entire function of bounded type in the upper half-plane. This extends the work of Holt and Vaaler (Duke Math J 83:203–247, 1996) for the signum function. We also provide periodic analogues of these results, finding optimal one-sided approximations by trigonometric polynomials of a given degree to a class of periodic functions with exponential subordination. These extremal trigonometric polynomials optimize the \(L^1({\mathbb {R}}/{\mathbb {Z}}, \mathrm{d}\vartheta )\)-error, where \(\vartheta \) is an arbitrary nontrivial measure on \({\mathbb {R}}/{\mathbb {Z}}\). The periodic results extend the work of Li and Vaaler (Indiana Univ Math J 48(1):183–236, 1999), who considered this problem for the sawtooth function with respect to Jacobi measures. Our techniques are based on the theory of reproducing kernel Hilbert spaces (of entire functions and of polynomials) and on the construction of suitable interpolations with nodes at the zeros of Laguerre–Pólya functions.

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Acknowledgments

E. C. acknowledges support from CNPq—Brazil Grants 302809/2011-2 and 477218/2013-0, and FAPERJ—Brazil Grant E-26/103.010/2012.

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Authors and Affiliations

  1. IMPA - Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina, 110, Rio de Janeiro, RJ, 22460-320, Brazil

    Emanuel Carneiro & Felipe Gonçalves

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  1. Emanuel Carneiro
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  2. Felipe Gonçalves
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Correspondence to Emanuel Carneiro.

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Carneiro, E., Gonçalves, F. Extremal problems in de Branges spaces: the case of truncated and odd functions. Math. Z. 280, 17–45 (2015). https://doi.org/10.1007/s00209-015-1411-1

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