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Let Π  n  d  denote the space of all spherical polynomials of degree at most n on the unit sphere  $$B(e,\alpha)=\big\{x\in\mathbb{S}^{d}:d(x,e)\leq\alpha\big\}\quad \bigl(e\in\mathbb{S}^{d},\ \alpha\in(0,\pi)\ \mbox{is bounded awayfrom}\ \pi\bigr),$$     we define the metric     $$\rho(x,y):=\frac{1}{\alpha}\sqrt{\bigl(d(x,y)\bigr)^{2}+\alpha\bigl(\sqrt{\alpha-d(x,e)}-\sqrt{\alpha-d(y,e)}\bigr)^{2}},$$     where x,y∈B(e,α). It is shown that given any β≥1, 1≤pΛ of B(e,α) satisfying the condition  $$\sum_{\omega\in \varLambda }\Bigl(\max_{x,y\in B_{\rho}(\omega,\,\beta\delta/n)}\big|f(x)-f(y)\big|^{p}\Bigr)\big|B_{\rho}(\omega ,\delta/n)\big|\leq (C\delta)^{p}\int_{B(e,\alpha)}\big|f(x)\big|^{p}\,d\sigma(x),$$     where d σ(x) denotes the Lebesgue measure on  $$B_{\rho}(x,r)=\bigl\{y\in B(e,\alpha):\rho(y,x)\leq r\bigr\}\quad (r>0),$$     and     $$\biggl|B_{\rho}\biggl(x,\frac{\delta}{n}\biggr)\biggr|=\int _{B_{\rho}(x,\delta/n)}d\sigma(y)\sim\alpha^{d}\biggl[\biggl(\frac{\delta}{n}\biggr)^{d+1}+\biggl(\frac{\delta}{n}\biggr)^{d}\sqrt{1-\frac {d(x,e)}{\alpha}}\biggr].$$     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.

Positive Cubature Formulas and Marcinkiewicz–Zygmund Inequalities on Spherical Caps