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Keywords:
entire functions; Paley-Wiener theorem; numerical interpolation; optimal interpolatory rule with prescribed nodes; remainder estimate
entire functions; Paley-Wiener theorem; numerical interpolation; optimal interpolatory rule with prescribed nodes; remainder estimate
Summary:
In the paper we present a derivative-free estimate of the remainder of an arbitrary interpolation rule on the class of entire functions which, moreover, belong to the space $L^2_{(-\infty ,+\infty )}$. The theory is based on the use of the Paley-Wiener theorem. The essential advantage of this method is the fact that the estimate of the remainder is formed by a product of two terms. The first term depends on the rule only while the second depends on the interpolated function only. The obtained estimate of the remainder of Lagrange’s rule shows the efficiency of the method of estimate. The first term of the estimate is a starting point for the construction of the optimal rule; only the optimal rule with prescribed nodes of the interpolatory rule is investigated. An example illustrates the developed theory.
References:
[1] I. S. Gradshtejn, I. M. Ryzhik: Tables of Integrals, Sums, Series and Products. Fizmatgiz, Moskva, 1963. (Russian)
[2] G. Hämmerlin: Über ableitungsfreie Schranken für Quadraturfehler II. Ergänzungen und Möglichkeiten zur Verbesserung. Numer. Math. 7 (1965), 232–237. DOI 10.1007/BF01436079 | MR 0184427
[3] W. Rudin: Real and Complex Analysis. McGraw-Hill, New York, 1966. MR 0210528 | Zbl 0142.01701
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