2017
Том 69
№ 9

# Multidimensional Lagrange–Yen-Type Interpolation Via Kotel'nikov–Shannon Sampling Formulas

Pogány T. K.

Abstract

Direct finite interpolation formulas are developed for the Paley–Wiener function spaces $L_\diamondsuit ^2$ and $L_{[-\pi, \pi]^d}^2$ , where $L_\diamondsuit ^2$ contains all bivariate entire functions whose Fourier spectrum is supported by the set ♦ = Cl{(u, v) ∣ |u| + |v| < π], while in $L_{[-\pi, \pi]^d}^2$ the Fourier spectrum support set of its d-variate entire elements is [−π, π] d . The multidimensional Kotel'nikov–Shannon sampling formula remains valid when only finitely many sampling knots are deviated from the uniform spacing. By using this interpolation procedure, we truncate a sampling sum to its irregularly sampled part. Upper bounds of the truncation error are obtained in both cases.

According to the Sun–Zhou extension of the Kadets $\frac{1}{4}$ -theorem, the magnitudes of deviations are limited coordinatewise to $\frac{1}{4}$ . To avoid this inconvenience, we introduce weighted Kotel'nikov–Shannon sampling sums. For $L_{[-\pi, \pi]^d}^2$ , Lagrange-type direct finite interpolation formulas are given. Finally, convergence-rate questions are discussed.

English version (Springer): Ukrainian Mathematical Journal 55 (2003), no. 11, pp 1810-1827.

Citation Example: Pogány T. K. Multidimensional Lagrange–Yen-Type Interpolation Via Kotel'nikov–Shannon Sampling Formulas // Ukr. Mat. Zh. - 2003. - 55, № 11. - pp. 1503-1520.

Full text