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

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.