Hermite Interpolation by Polynomials and Trigonometric Polynomials

Let α1, . . . , αn be any set of n positive integers. The problem is to find the smallest N such that, for any set of n distinct points z1, . . . , zn in the complex plane and α1 + · · · + αn arbitrarily prescribed values

w0,ν , . . . , wαν−1,ν (ν = 1, . . . , n) ,

there exists a polynomial p(z) := ∑N

ν=0 aν z ν of degree not exceeding

N satisfying

p(j)(z1) = wj,1 (j = 0, . . . , α1 − 1) . . .