Construction of interval designs for Gegenbauer measure

Construct interval designs for the Gegenbauer measure $(1-t^2)^{\lambda-1/2}dt$ for general $\lambda>-1/2$, in order to support product-rule constructions of spherical designs.

Background

The paper explains that many known product rules for spherical designs depend on the existence of interval designs associated with Gegenbauer measures. Although general existence results for spherical designs are known, the corresponding interval-design construction problem remains unresolved and limits the applicability of these product rules.