On Optimal Cardinal Interpolation
For the Hardy classes of functions analytic in the strip around real axis of a size 2β, an optimal method of cardinal interpolation has been proposed within the framework of Optimal Recovery . 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 β → ∞, the optimal interpolant converges to the well-knownNyquist–Shannon cardinal series.
Keywordscardinal interpolation Optimal Recovery Hardy classes Jacobi elliptic functions infinite Blaschke product interference effect sinc filter
2000 Mathematics Subject Classification62G08 secondary 62K05 33E05 42A15
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