Problems of Chebyshev quadrature on sphere and circle

The principal problem is to find optimal or nearly optimal $N$-tuples of nodes for Chebyshev quadrature on the unit sphere when $N$ is large. One tries to obtain such nodes as solutions to suitable extremalproblems. In earlier work the author has described a complex-analytic method which would show that the extremal $N$-tuples provide good Chebyshev nodes.These are nodes such that, for constants $A,\,b,\,c>0$, the quadrature remainder is $\le Ae^{-b\sqrt N}$ for the polynomials of degree $\le c\sqrt N$ and sup norm$1$. However, in order to apply this method it is necessary to establish uniform distribution and good separation of the extremal points. In this note uniform distribution is proved with the aid of potential theory. It is plausible that one also has adequate separation.Indeed, there is such separation in the corresponding extremal problems for the unit circle which are considered at length.