We consider convergence sets of formal power series f(z,t)=∞∑n=0 fn(z)tn,where fn(z)are holomorphic functions on a domain Ω in C.A subset E of Ω is said to be a convergence set in Ω if there is a series f(z,t)such that E is exactly the set of points z for which f(z,t)converges as a power series in a single variable t in some neighborhood of the origin.A σ-convex set is defined to be the union of a countable collection of polynomially convex compact subsets.We prove that a subset of C is a convergence set if and only if it is σ-convex.

The identification of compact surfaces of genus-0 is discussed.First,the algorithm of the conformal mapping from the surface to sphere is given.Second,the harmonic expansion of the conformal mapping is computed to realize the identification,compression and reconstruction of 3D surface.

In the present paper, both the perfect convergence for the Lagrange interpolation of analytic functions on [ − 1, 1] and the perfect convergence for the trigono-metric interpolation of analytic functions on [ − p, p] with period 2p are discussed.