Optimal Truncations for Multivariate Fourier and Chebyshev Series: Mysteries of the Hyperbolic Cross: Part I: Bivariate Case

Abstract

The key to most successful applications of Chebyshev and Fourier spectral methods in high space dimension are a combination of a Smolyak sparse grid together with so-called “hyperbolic cross” truncation. It is easy to find counterexamples for which the hyperbolic cross truncation is far from optimal. An important question is: what characteristics of a function make it “crossy”, that is, suitable for the hyperbolic crosss truncation? We have not been able to find a complete answer to this question. However, by combining low-rank SVD approximation, Poisson summation and imbricate series, hyperbolic coordinates and numerical experimentation, we are, to borrow from Fermi, “confused at a higher level”. For rank-one (separable) functions, which are the product of two univaraiate functions, we show that the hyperbolic cross truncation is indeed the best if the functions have weak singularities on the domain or boundaries so that the spectral series has a finite order of power-law convergence. For functions smooth on the domain, and therefore blessed with exponentially convergent spectral series, we have failed to find any reasonable examples where the hyperbolic cross truncation is best.