Multivariate polynomial interpolation on Lissajous–Chebyshev nodes

Rhodonea Curves as Sampling Trajectories for Spectral Interpolation on the Unit Disk

“…Rhodonea curves can be regarded as polar counterparts of bivariate Lissajous curves on the square [−1, 1] 2 and of spherical Lissajous curves. This article is a continuation of the work on polynomial interpolation on Lissajous curves [7,8,11,13] and on spherical Lissajous nodes [12] and extends it to the polar setting. The differences between the actual work on the disk and the previous works on the hypercube and the unit sphere arise naturally from the differing geometries.…”

“…Nevertheless, the core ideas in all three setups are similar and many of the ideas used for Lissajous curves can be carried over to the setting of rhodonea curves. In particular, as for multivariate Lissajous-Chebyshev points in the hypercube [7,8], a main step in the proof of the quadrature and interpolation formulas is a discrete orthogonality structure linked to the structure of the rhodonea nodes. Compared to previous works, a major progress in this article is the larger flexibility in the choice of the interpolation space.…”

“…In this case, the black and white dots indicate the two interlacing grids determining RD The frequency parameters m 1 and m 2 in the curve (m) α determine a superposition of a radial and an angular harmonic motion. For this reason, rose curves can also be regarded as polar variants of bivariate Lissajous curves [7,8,11,13]. If the numbers m 1 and m 2 are relatively prime, the minimal period P of (m) α is given by P = 2π if m 1 + m 2 is odd, and P = π if m 1 + m 2 is even (see Proposition 1).…”

“…Interpolation spaces with such an averaging for some of the boundary elements of the spectral index set were originally studied in the bivariate setting for the Morrow-Patterson-Xu points in [20,36]. For multivariate interpolation on Lissajous-Chebyshev nodes, this averaging process is studied in more detail in [7].…”

A spectral interpolation scheme on the unit sphere based on the nodes of spherical Lissajous curves

“…with a frequency vector m = (m 1 , m 2 ) ∈ N 2 and a rotation parameter α ∈ R. The curve (m) α lies in the unit sphere S 2 = {x ∈ R 3 : x 2 1 + x 2 2 + x 2 3 = 1} of the three-dimensional space R 3 . Similar as for bivariate Lissajous curves [7,8,11,12], the curve (m) α describes a superposition of a latitudinal and a longitudinal harmonic motion determined by the frequencies m 1 and m 2 .…”