The Constant of Interpolation

We prove that a suitably adjusted version of Peter Jones’ formula for interpolation in H ∞ gives a sharp upper bound for what is known as the constant of interpolation. We show how this leads to precise and computable numerical bounds for this constant. With each finite or infinite sequence Z = (zj) (j = 1, 2,...) of distinct points zj = xj+iyj in the upper half-plane of the complex plane, we associate a number M(Z) ∈ R+ ∪ {+∞} which we call the constant of interpolation. We may define it in two equivalent ways. The first is related to Carleson’s interpolation theorem for H ∞ [Car58]. We say that Z is an interpolating sequence if the interpolation problem f(zj) = wj, j = 1, 2,...(1) has a solution f ∈ H ∞ for each bounded sequence (wj) of complex numbers. Using the open mapping theorem, we find that if Z is an interpolating sequence, then we can always solve (1) with a function f such that ‖f‖ ∞ ≤ C‖(wj)‖∞ for some C < ∞ depending only on Z. The constant of interpolation M(Z) is declared to be the smallest such C. We set M(Z) = + ∞ if Z is not an interpolating sequence. By a classical theorem of Pick (see [Gar81, p. 2]), we may alternatively define M(Z) as follows: Let Mn(Z) be the smallest number C such that the matrices ( 1 − wjwk zj − zk j,k=1,2,...,n are positive semi-definite whenever ‖(wj)‖ ∞ ≤ 1/C. The constant of inter-polation is then M(Z) = Mn(Z) if Z is a finite sequence consisting of n points and M(Z) = limn→∞Mn(Z) if Z is infinite. We will make no use of this definition, but have stated it to make the reader aware of the rel-evance of M(Z) for the classical Nevanlinna-Pick problem. Note that this connection has been investigated by Koosis, who derived Carleson’s theorem formally from Pick’s theorem [Koo00]