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On optimal cardinal interpolation. (English) Zbl 1417.62081

Summary: For the Hardy classes of functions analytic in the strip around real axis of a size \(2\beta\), an optimal method of cardinal interpolation has been proposed within the framework of optimal recovery [B. Levit, Math. Methods Stat. 25, No. 4, 235–261 (2016; Zbl 1362.62090)]. Below this method, based on the Jacobi elliptic functions, is shown to be optimal according to the criteria of nonparametric regression and optimal design.
In a stochastic non-asymptotic setting, the maximal mean squared error of the optimal interpolant is evaluated explicitly, for all noise levels away from 0. A pivotal role is played by the interference effect, in which the oscillations exhibited by the interpolant’s bias and variance mutually cancel each other. In the limiting case \(\beta \rightarrow\infty\), the optimal interpolant converges to the well-known Nyquist-Shannon cardinal series.

MSC:

62G08 Nonparametric regression and quantile regression
62K05 Optimal statistical designs
33E05 Elliptic functions and integrals
42A15 Trigonometric interpolation

Citations:

Zbl 1362.62090
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References:

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