  Approximation on the spherical cap is different from that on the sphere which
requires us to construct new operators. This paper discusses the approximation
on the spherical cap. That is, so called Jackson-type operator
$\{J_{k,s}^m\}_{k=1}^{\infty}$ is constructed to approximate the function
defined on the spherical cap $D(x_0,\gamma)$. We thus establish the direct and
inverse inequalities and obtain saturation theorems for
$\{J_{k,s}^m\}_{k=1}^{\infty}$ on the cap $D(x_0,\gamma)$. Using methods of
$K$-functional and multiplier, we obtain the inequality \begin{eqnarray*}
C_1\:\| J_{k,s}^m(f)-f\|_{D,p}\leq \omega^2\left(f,\:k^{-1}\right)_{D,p} \leq
C_2 \max_{v\geq k}\| J_{v,s}^m(f) - f\|_{D,p} \end{eqnarray*} and that the
saturation order of these operators is $O(k^{-2})$, where
$\omega^2\left(f,\:t\right)_{D,p}$ is the modulus of smoothness of degree 2,
the constants $C_1$ and $C_2$ are independent of $k$ and $f$.

Classical Analysis and ODEs