Polynomial approximation and quadrature on geographic rectangles

Abstract

Using some recent results on subperiodic trigonometric interpolation and quadrature, and the theory of admissible meshes for multivariate polynomial approximation, we study product Gaussian quadrature, hyperinterpolation and interpolation on some regions of dS,d ≥ 2. Such regions include caps, zones, slices and more generally spherical rectangles defined on S2 by longitude and (co)latitude (geographic rectangles). We provide the corresponding Matlab codes and discuss several numerical examples on S

Numerical hyperinterpolation over nonstandard planar regions

“…For example, one might think applying the method to latitude-longitude rectangles of the sphere (with spherical caps as a special case, cf. [17]), using a basis of (n + 1) 2 spherical harmonics. The latter, however, tends to be very ill-conditioned already at small degrees, since spherical harmonics are orthogonal and thus well-conditioned only on the whole sphere.…”

“…On the other hand, several positive cubature formulas based on product Gaussian quadrature have been obtained for the ordinary area or surface measure on nonstandard geometries, via suitable geometric transformations; cf., e.g., [7,8,17,21,27]. We present here some orthogonalization examples that exploit such product type formulas.…”

Subperiodic Dubiner distance, norming meshes and trigonometric polynomial optimization

“…See Figure 1 for two examples on the sphere and on the torus. More generally, working by the appropriate geometric transformation and coordinates we can apply the discrete optimization method on standard sections of disk, sphere and ball, such as caps, lenses, lunes, sectors, slices; see [11,16] for several instances of this kind with the relevant geometric transformations. The corresponding polynomial meshes have constant C = C 2 * (planar and surface instances) or C = C 3 * (solid instances), and cardinality O((mn) 2 ) or O((mn) 3 ), respectively.…”

Subperiodic trigonometric subsampling: A numerical approach

“…In recent years some attention has been devoted in the numerical literature to trigonometric approximation on subintervals of the period. Relevant topics are the theory of Fourier extensions, where one of the main initial motivations was of circumventing the Gibbs phenomenon (cf., e.g., [1,2,7]), and the theory of subperiodic interpolation and quadrature, whose main motivation came from polynomial approximation on domains related to circular/elliptical arcs, such as sections of disk and sphere (cf., e.g., [6,8,10,16]).…”