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Spectral polynomial approximation of smooth functions allows real-timemanipulation of and computation with them, as in the Chebfun system. Extensionof the technique to two-dimensional and three-dimensional functions onhyperrectangles has mainly focused on low-rank approximation. While this methodis very effective for some functions, it is highly anisotropic and unacceptablyslow for many functions of potential interest. A method based on automaticrecursive domain splitting, with a partition of unity to define the globalapproximation, is easy to construct and manipulate. Experiments show it to beas fast as existing software for many low-rank functions, and much faster onother examples, even in serial computation. It is also much less sensitive toalignment with coordinate axes. Some steps are also taken toward approximationof functions on nonrectangular domains, by using least-squares polynomialapproximations in a manner similar to Fourier extension methods, with promisingresults.

An adaptive partition of unity method for multivariate Chebyshev polynomial approximations