Positive Cubature Formulas and Marcinkiewicz–Zygmund Inequalities on Spherical Caps

Let Π n denote the space of all spherical polynomials of degree at most n on the unit sphere S of Rd+1, and let d(x, y) denote the geodesic distance arccosx ·y between x, y ∈ S . Given a spherical cap B(e,α)= x ∈ S : d(x, e)≤ α e ∈ S , α ∈ (0,π) is bounded away from π, we define the metric ρ(x, y) := 1 α √( d(x, y) )2 + αα − d(x, e)−α− d(y, e)2, where x, y ∈ B(e,α). It is shown that given any β ≥ 1, 1 ≤ p < ∞ and any finite subset Λ of B(e,α) satisfying the condition min ξ,η∈Λ ξ =η ρ(ξ, η) ≥ δ n with δ ∈ (0,1], there exists a positive constant C, independent of α, n, Λ and δ, such that, for any Communicated by Doron S. Lubinsky. Research started while the second author visited Edmonton. The first author was partially supported by the NSERC Canada under grant G121211001. The second author was partially supported by the Beijing Natural Science Foundation (1062004), by the National Natural Science Foundation of China (10871132), and by a grant from the Key Programs of Beijing Municipal Education Commission (KZ200810028013). F. Dai Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada e-mail: [email protected] H. Wang ( ) School of Mathematics Science, Capital Normal University, Beijing 100048, People’s Republic of China e-mail: [email protected] 2 Constr Approx (2010) 31: 1–36 f ∈Πd n , ∑ ω∈Λ ( max x,y∈Bρ(ω,βδ/n) ∣f (x)− f (y)∣p )∣Bρ(ω, δ/n) ∣∣ ≤ (Cδ) ∫ B(e,α) ∣f (x) ∣∣p dσ(x), where dσ(x) denotes the Lebesgue measure on S , Bρ(x, r)= { y ∈ B(e,α) : ρ(y, x)≤ r (r > 0), and ∣∣∣∣Bρ ( x, δ n )∣∣∣∣= ∫ Bρ(x,δ/n) dσ (y)∼ α [( δ n )d+1 + ( δ n )d√ 1 − d(x, e) α ] . As a consequence, we establish positive cubature formulas and Marcinkiewicz– Zygmund inequalities on the spherical cap B(e,α). Moreover, a higher-dimensional analogue of the large sieve inequality of Golinskii, Lubinsky, and Nevai (J. Number Theory 91(2):206–229, 2001) is obtained.