Multidimensional Lagrange-Yen-type interpolation via Kotel'nikov-Shannon sampling formulas

Direct finite interpolation formulas are developed for the Paley-Wiener function spaces $L^2_\Diamond$ and $L^2_{; ; [-\pi, \pi]^d}; ; $, where $L^2_\Diamond$ contains all bivariate functions whose Fourier spectrum is supported by the set $\Diamond = {; ; \rm Cl}; ; \{; ; (u, v)| |u|+|v|< \pi\}; ; $, while in $L^2_{; ; [-\pi, \pi]^d}; ; $ the Fourier spectrum support set of its $d$-variate entire elements is $[-\pi, \pi]^d$. The multidimensional Kotel'nikov - Shannon sampling formula reamins valid when only finitely many sampling knots are deviated from the uniform sampling. 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 1/4 - theorem, the magnitude of deviations are limited coordinatewise to 1/4. To avoid this inconvenience, we introduce the weighted Kotel/nikov - Shannon sampling sums. For $L^2_{; ; [-\pi, \pi]^d}; ; $ Lagrange-type direct finite interpolation fromulas are given. Finally, convergence - rate questions are discussed.