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Distribution of nodes on algebraic curves in N . (English) Zbl 1044.32026
Let A N be an irreducible curve and let KA be a non-polar compact set. For d=0,1,, let m d be the dimension of the space P d |A of all complex polynomials of degree at most d restricted to A. It is known that for sufficiently large d we have m d =dD+c, where D is the degree of A and c is an integer. Consider Lagrange interpolation polynomials L d f(z)= j=1 m d f(A d,j ) j (d) (z) with nodes A d,j K. Let 𝛬 d := j=1 m d | j (d) | K . Assume that the nodes are chosen in such a way that lim sup d+ 𝛬 d 1/d 1. Then (1/m d ) j=1 m d δ A d,j weak--*(1/2πD)dd c V K * =:μ K and suppμ K K, where V K is the Siciak extremal function for K.
MSC:
32U05Plurisubharmonic functions and generalizations
31C10Pluriharmonic and plurisubharmonic functions
41A05Interpolation (approximations and expansions)