Approximation on non-tensor domains including squircles, Part III: Polynomial hyperinterpolation and radial basis function interpolation on Chebyshev-like grids and truncated uniform grids

Single-domain spectral methods have been largely restricted to tensor product bases on a tensor product grid. To break the “tensor barrier”, we study approximation in a domain bounded by a “squircle”, the zero isoline of B(x,y)=x2ν+y2ν−1. The boundary varies smoothly from a circle [ν=1] to the square [ν=∞]. Polynomial least-squares hyperinterpolation converges geometrically as long as the number of points P is (at least) double the number of basis functions N. The polynomial grid was made denser near the boundaries (“Chebyshev-like”) by depositing grid points along wisely chosen contours of B. Gaussian radial basis functions (RBFs) were more robust in the sense that they, too, converged geometrically, but hyperinterpolation (P>N) and a Chebyshevized grid were unnecessary. A uniform grid, truncated to include only those points within the squircle, was satisfactory even without interpolation points on the boundary (although boundary points are a cost-effective improvement). For a given number of points P, however, RBF interpolation was only slightly more accurate than polynomial hyperinterpolation, and needed twice as many basis functions. Interpolation costs can be greatly reduced by exploiting the invariance of the squircle-bounded domain to the eight elements D4 dihedral group.