APPLICATIONS OF MATHEMATICS, Vol. 46, No. 6, pp. 401-417, 2001

# Interpolation formulas for functions of exponential type

## Josef Kofron, Emilie Moravcova

Josef Kofron, Charles University, Faculty of Mathematics and Physics, Department of Numerical Mathematics, Sokolovska 83, 186 75 Praha 8, Czech Republic, e-mail: kofron@karlin.mff.cuni.cz; Emilie Moravcova, AQUILA TS s.r.o., Vaclavske namesti 76, 561 51 Letohrad, Czech Republic, e-mail: aquila@aquila.cz

Abstract: 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.

Keywords: entire functions, Paley-Wiener theorem, numerical interpolation, optimal interpolatory rule with prescribed nodes, remainder estimate

Classification (MSC 2000): 65D05, 41A05, 41A80

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