  A spherical $t$-design is a set of points on the sphere that are nodes of a
positive equal weight quadrature rule having algebraic accuracy $t$ for all
spherical polynomials with degrees $\le t$. Spherical $t$-designs have many
distinguished properties in approximations on the sphere and receive remarkable
attention. Although the existence of a spherical $t$-design is known for any
$t\ge 0$, a spherical design is only known in a set of interval enclosures on
the sphere \cite{chen2011computational} for $t\le 100$. It is unknown how to
choose a set of points from the set of interval enclosures to obtain a
spherical $t$-design. In this paper we investigate a new concept of point sets
on the sphere named spherical $t_\epsilon$-design ($0<\epsilon<1$), which are
nodes of a positive weight quadrature rule with algebraic accuracy $t$. The sum
of the weights is equal to the area of the sphere and the mean value of the
weights is equal to the weight of the quadrature rule defined by the spherical
$t$-design. A spherical $t_\epsilon$-design is a spherical $t$-design when
$\epsilon=0,$ and a spherical $t$-design is a spherical $t_\epsilon$-design for
any $0<\epsilon <1$. We show that any point set chosen from the set of interval
enclosures \cite{chen2011computational} is a spherical $t_\epsilon$-design. We
then study the worst-case errors of quadrature rules using spherical
$t_\epsilon$-designs in a Sobolev space, and investigate a model of polynomial
approximation with the $l_1$-regularization using spherical
$t_\epsilon$-designs. Numerical results illustrate good performance of
spherical $t_\epsilon$-designs for numerical integration and function
approximation on the sphere.
