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Interpolation for solutions of the Helmholtz equation
Author(s) -
González–Casanova Pedro,
Wolf Kurt Bernardo
Publication year - 1995
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.1690110107
Subject(s) - mathematics , sinc function , helmholtz equation , mathematical analysis , interpolation (computer graphics) , fourier transform , helmholtz free energy , sampling (signal processing) , dimension (graph theory) , pure mathematics , physics , computer graphics (images) , quantum mechanics , computer science , boundary value problem , filter (signal processing) , computer vision , animation
We study the interpolation problem for solutions of the two‐dimensional Helmholtz equation, which are sampled along a line. The data are the function values and the normal derivatives at a discrete set of point sensors. A wave transform is used, analogous to the common Fourier transform. The inverse wave transform defines the Hilbert space for oscillatory Helmholtz solutions. We thereby introduce an interpolant that has some advantages over the usual sinc x in the Whittaker–Shannon sampling in one dimension; in particular, coefficients of the two‐dimensional solution are invariant under translations and rotations of the sampling line. The analysis is relevant for the optical sampling problem by sensors on a screen. © 1995 John Wiley & Sons, Inc.