## 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

Zbl 1362.62090
Full Text:

### References:

 [1] N. I. Akhiezer, Elements of the Theory of Elliptic Functions (AMS, Providence, R.I., 1970). [2] Artiles, L.; Levit, B., Adaptive Regression on the Real Line in Classes of Smooth Functions, Austrian J. Statist., 32, 99-129, (2003) [3] H. Bateman and A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1955), Vol. 2. · Zbl 0143.29202 [4] R. P. Boas, Entire Functions (Academic Press, New York, 1954). · Zbl 0058.30201 [5] Cho, J.; Levit, B., Asymptotic Optimality of Periodic Spline Interpolation in Nonparametric Regression, J. Statist. Theory Pract., 2, 465-474, (2008) · Zbl 1211.62063 [6] Golubev, Y.; Levit, B.; Tsybakov, A., Asymptotically Efficient Estimation of Analytic Functions in Gaussian Noise, Bernoulli, 2, 167-181, (1996) · Zbl 0860.62034 [7] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, New York, 1994), 5th ed. · Zbl 0918.65002 [8] Ibragimov, I., Estimation of Analytic Functions, IMS Lecture Notes Monogr. Ser., 36, 359-383, (2001) · Zbl 1373.62130 [9] B. Ya. Levin, Yu. Lyubarski, M. Sodin, and V. Tkachenko, Lectures on Entire Functions (AMS, Providence, RI, 1996). [10] Levit, B., Some New Perspectives in Best Approximation and Interpolation of Random Data, Math. Meth. Statist., 22, 165-192, (2013) · Zbl 1294.62169 [11] Levit, B., Variance-Optimal Data Approximation Using the Abel-Jacobi Functions, Math. Meth. Statist., 24, 37-54, (2015) · Zbl 1319.62155 [12] Levit, B., Optimal Methods of Interpolation in Nonparametric Regression, Math. Meth. Statist., 25, 235-261, (2016) · Zbl 1362.62090 [13] A. I.Markushevich, The Theory of Functions of a Complex Variable (Chelsea Publ., New York, 1985). [14] K. Yu. Osipenko, Optimal Recovery of Analytic Functions (Nova Science Publ., New York, 2000). [15] A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977). · Zbl 0422.94001 [16] W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1987), 3d ed. · Zbl 0925.00005 [17] I. J. Schoenberg, Cardinal Spline Interpolation (SIAM, Philadelphia, PA, 1973). · Zbl 0264.41003 [18] A. N. Shiryaev, Probability-1, in Graduate Texts in Mathematics (Springer, New York, 2016), 3rd ed.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.