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Positive Cubature formulas and Marcinkiewicz-Zygmund inequalities on spherical caps | | Feng Dai
; Heping Wang
; | | Date: |
26 Mar 2007 | | Subject: | Classical Analysis and ODEs; Numerical Analysis | | 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 $$ B(e,al)={xinsph: d(x, e) leq al}, (einsph, ext{$alin (0,pi)$ is bounded away from $pi$}),$$ we define the metric $$
ho(x,y):=frac 1{al} sqrt{(d(x, y))^2+al(sqrt{al-d(x, e)}-sqrt{al-d(y,e)})^2},
$$ where $x, yin B(e,al)$. It is shown that given any $ege 1$, $1leq p<infty$ and any finite subset $Ld$ of $B(e,al)$ satisfying the condition $dmin_{sub{xi,eta in Ld xi
eq eta}}
ho (xi,eta) ge f da n$ with $dain (0,1]$, there exists a positive constant $C$, independent of $al$, $n$, $Ld$ and $da$, such that, for any $finPi_{n}^d$, egin{equation*} sum_{ogin Ld} (max_{x,yin B_
ho (og, eda/n)} f(x)-f(y) ^p) B_
ho(og, da/n) le (C dz)^p int_{B(e,al)} f(x) ^p dsa(x),end{equation*} where $dsa(x)$ denotes the usual Lebesgue measure on $sph$, $$B_
ho(x, r)=Bl{yin B(e,al):
ho(y,x)leq rBr}, (r>0),$$ and $$Bl B_
ho(x, fda n)Br =int_{B_{
ho}(x, da/n)} dsa(y) sim al ^{d}Bl[ (f{da}n)^{d+1}+ (fda n)^{d} sqrt{1-f{d(x, e)}al}Br].$$ As a consequence, we establish positive cubature formulas and Marcinkiewicz-Zygmund inequalities on the spherical cap $B(e,al)$. | | Source: | arXiv, math/0703768 | | Services: | Forum | Review | PDF | Favorites | |
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