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- W2952993265 abstract "Let $Pi_n^d$ denote the space of all spherical polynomials of degree at most $n$ on the unit sphere $sph$ of $mathbb{R}^{d+1}$, and let $d(x, y)$ denote the usual geodesic distance $arccos xcdot y$ between $x, yin sph$. Given a spherical cap $$ B(e,al)={xinsph: d(x, e) leq al}, (einsph, text{$alin (0,pi)$ is bounded away from $pi$}),$$ we define the metric $$rho(x,y):=frac 1{al} sqrt{(d(x, y))^2+al(sqrt{al-d(x, e)}-sqrt{al-d(y,e)})^2}, $$ where $x, yin B(e,al)$. It is shown that given any $bege 1$, $1leq p 0),$$ and $$Bl|B_rho(x, fda n)Br|=int_{B_{rho}(x, da/n)} dsa(y) sim al ^{d}Bl[ (f{da}n)^{d+1}+ (fda n)^{d} sqrt{1-f{d(x, e)}al}Br].$$ As a consequence, we establish positive cubature formulas and Marcinkiewicz-Zygmund inequalities on the spherical cap $B(e,al)$." @default.
- W2952993265 created "2019-06-27" @default.
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- W2952993265 date "2007-03-26" @default.
- W2952993265 modified "2023-09-25" @default.
- W2952993265 title "Positive Cubature formulas and Marcinkiewicz-Zygmund inequalities on spherical caps" @default.
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