Abstract
Let \(E= A-iB\) be a Hermite–Biehler entire function of exponential type \(\tau /2\) where A and B are real entire, and consider \(\mathrm{d}\mu (x) = \mathrm{d}x/|E(x)|^2\). We show that the sign of the product AB is an extremal signature for the space of functions of exponential type \(\tau \) with respect to the norm of \(L^1(\mu )\). This allows us to find best approximations by entire functions of exponential type \(\tau \) in \(L^1(\mu )\)-norm to certain special functions (e.g., the Gaussian and the Poisson kernel).
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Communicated by Edward B. Saff.
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Littmann, F., Spanier, M. Extremal Signatures. Constr Approx 47, 339–356 (2018). https://doi.org/10.1007/s00365-017-9373-7
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DOI: https://doi.org/10.1007/s00365-017-9373-7
Keywords
- Best approximation
- Extremal signature
- Bandlimited function
- Exponential type
- Hardy space
- Hermite–Biehler function