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. 0 1995 John Wiley & Sons, Inc.

In this work, by using the local node refinement technique purposed in [1,2], and a quad-tree type algorithm [3,4]

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During the last years, there has been increased interest in developing efficient radial basis function (RBF) algorithms to solve partial differential problems of great scale. In this article, we are interested in solving large PDEs problems, whose solution presents rapid variations. Our main objective is to introduce a RBF dynamical domain decomposition algorithm which simultaneously performs a node adaptive strategy. This algorithm is based on the RBFs unsymmetric collocation setting. Numerical experiments performed with the multiquadric kernel function, for two stationary problems in two dimensions are presented.

Modeling and contouring of geophysical data often require distributions of regularly spaced values. Splines have been shown to be the most accurate methods to obtain such distributions. We emphasize the general problem of interpolating random distributions of data on a given surface. Splines are classified into unidimensional, quasi‐bidimensional, and strictly bidimensional; based on this classification, a systematic derivation of the corresponding interpolating techniques is conducted. Two approaches are presented to obtain unidimensional splines: one based on the continuity of the first and second derivatives of the polynomials involved, and the other based on a variational approach. Quasi‐bidimensional splines are constructed based on the unidimensional approach, while strictly bidimensional splines are generated by minimizing the bidimensional curvature. Quasi‐bidimensional splines can be used for processing data distributions along nearly parallel lines; linear projections and parameterization are the techniques used in interpolating this type of distribution. Strictly bidimensional splines minimize curvature through the analytic solution of the Euler‐Lagrange equation or by a finite‐difference algorithm. The maximum error, mean error, and standard deviation between interpolated data and exact field values produced by various prisms show that quasi‐bidimensional splines are 2.7 percent more accurate in the maximum error than strictly bidimensional splines when both techniques are applied to regularly spaced data. However, for irregularly spaced data, three examples containing 300, 600, and 900 random data points show the superiority of the thin‐plate approach over the quasi‐bidimensional splines. A comparison between various interpolation densities on regular grids, starting from a set of 327 randomly distributed magnetic stations, illustrates some differences between geophysically meaningful interpolations and interpolations carried out only for contouring purposes.