Optimal cubature formulas on compact homogeneous manifolds

Abstract

We find lower bounds for the rate of convergence of optimal cubature formulas on sets of differentiable functions on compact homogeneous manifolds of rank I or two-point homogeneous spaces. It is shown that these lower bounds are sharp in the power scale in the case of S 2 , the unit sphere in R 3 .

Quadrature rules and distribution of points on manifolds

“…In particular, (B) and (D) for p = 2 and for spheres are contained in [7,[17][18][19] and [20]. For Besov spaces on spheres some results slightly more precise than (B) and (D) are in [21], while a result slightly weaker than (D) for compact two point homogeneous spaces is in [25]. See also [10] and, for a survey on related results, [15] and [30].…”

Approximation on the complex sphere

“…where the penultimate and ultimate steps are justified by the condition Cn ≤ l < n (see, e.g., [14]). Consequently, r Cn,n−Cn,n ≤ C for any n ∈ N and using (23) we get ϑ m ≥ C p 1/2 • 2 C(q ′ ) 1/2 , p < ∞, q > 1, (log m) 1/2 • 2 C(q ′ ) 1/2 , p = ∞, q > 1,…”

Quadrature rules and distribution of points on manifolds

“…In particular, (3) and (4) for p = 2 and for spheres are contained in [7], [15] and [16]. For Besov spaces on spheres a result slightly more precise than (3) is in [17], while a result slightly weaker than (4) for compact two point homogeneous spaces is in [21]. See also [10] and, for a survey on related results, [14] and [19].…”