I am interested in some sort of analytic interpolation. A toy version of my problem is as follows. Let $V \subset \mathbb{C}$ be a complex neighborhood of $[0,1]$. Assume there is some bounded nonconstant analytic function $v_0(x) : V \to \mathbb{C}$, that further satisfies $v_0(x) \geq 0$ for all $x \in [0,1]$. For $h_k = 2^{-k}$, $k=0,1,2,\ldots$ partition the interval $[0,1]$ in the natural way $$0 = x_0 < x_1 < \ldots < x_{2^k} = 1 \text{ with } x_j = jh_k.$$ Assume that $v_{h_k}(x) : [0,1] \to [0,\infty)$ is constant on each interval $x \in [x_j,x_{j+1})$ (i.e. a piecewise constant function) and that $\sup_x |v_{h_k}(x) - v_{0}(x)| < L|h_k|$ for some Lipschitz constant $L<\infty$.
I am looking for a function $\Phi(x,h) : [0,1] \times [0,1] \to [0,\infty)$ and a neighborhood $V' \subset V \subset \mathbb{C}$ of $[0,1]$ with the following properties.
- $\sup_{x \in V'} |\Phi(x,h)-\Phi(x,h')| < L' |h-h'|$ for all $0 \leq h,h' \leq 1$.
- For each $0 \leq h \leq 1$, $x \to \Phi(x,h)$ is analytic on $V'$ and $\Phi([0,1],h) \geq 0$.
- For $k=0,1,2,\ldots$ and $j=1,\ldots,2^k$ and $x \in [x_{j-1},x_j)$, $$v_{h_k}(x) = {1 \over h_k} \int_{x_j}^{x_{j+1}} \Phi(x,h_k) \, dx \qquad \text{ ("quasi interpolation")}$$
I also want $\Phi(x,0) = v_0(x)$ on $[0,1]$, but I hope this is implied by 1.--3.
Are my hypotheses sufficient? I'm aware of the interpolation theorems of Nevanlinna and Carleson but I wasn't able to make those work for me, I'm not even sure if that's the right approach.
