Multidimensional Lagrange–Yen-Type Interpolation Via Kotel’nikov–Shannon Sampling Formulas - Pogány, T. K. (Ukrains’kyi Matematychnyi Zhurnal, 2003)
Direct finite interpolation formulas are developed for the Paley–Wiener function spaces \(L_^2\) and \(L_[-π, π]^d^2\) , where \(L_^2\) contains all bivariate entire functions whose Fourier spectrum is supported by the set ♦ = Cl(u, v) ∣ |u| + |v| < π], while in \(L_[-π, π]^d^2\) 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 \(14\) -theorem, the magnitudes of deviations are limited coordinatewise to \(14\) . To avoid this inconvenience, we introduce weighted Kotel’nikov–Shannon sampling sums. For \(L_[-π, π]^d^2\) , Lagrange-type direct finite interpolation formulas are given. Finally, convergence-rate questions are discussed.