**Background**: Let $S$ denote the so-called Schur class of complex analytic functions from the open unit disk $D$ in $\mathbb{C}$ to the closed unit disk $\overline{D}$. Given distinct points $z_1,\ldots,z_n\in D$ and points $w_1,\ldots,w_n\in\overline{D}$, G. Pick's Theorem (1916) says that there is an $f\in S$ such that $f(z_j)=w_j$ for each $j$ if and only if the matrix $\left(\frac{1-w_j\overline{w_k}}{1-z_j\overline{z_k}}\right)_{j,k=1}^n$ is positive semidefinite. Suppose that $g:D\to\mathbb{C}$ is a function. By Pick's Theorem it follows immediately that a necessary condition for $g$ to be in $S$ is that for each $n$ and each $n$-tuple of distinct points $z_1,\ldots,z_n\in D$, the matrix $\left(\frac{1-g(z_j)\overline{g(z_k)}}{1-z_j\overline{z_k}}\right)_{j,k=1}^n$ is positive semidefinite. I find it remarkable that this is also a sufficient condition (this too can be derived using Pick's Theorem, although there is a more general method, as indicated in Edit 2 and Yemon Choi's answer). However, I do not know whether Pick observed this fact, and I don't see it in I. Schur's related work from around that time (1917-1918).

**Question**: Who first observed that a function $g:D\to\overline{D}$ is analytic if for each $n$ and each $n$-tuple of distinct points $z_1,\ldots,z_n\in D$, the matrix $\left(\frac{1-g(z_j)\overline{g(z_k)}}{1-z_j\overline{z_k}}\right)_{j,k=1}^n$ is positive semidefinite?

To elaborate a little, the theory of reproducing kernel Hilbert spaces and their multipliers in which this characterization can now be found was developed well after people including Pick, Nevanlinna, and Schur had developed other aspects of the theory of bounded analytic functions on the unit disk. Part of the reason I would be interested in knowing the origin is that it would be interesting to see if completely different methods were used. If different methods weren't used, learning of the origin of this would help me to better understand the history of the uses of Hilbert space in function theory.

**Edit 1**: In an attempt to increase clarity, I changed the codomain from $\mathbb{C}$ to $\overline{D}$, so that the trivial condition of being bounded by 1 is no longer emphasized.

**Edit 2**: I'm still curious about the history of this, but I've realized that I was confused about the logical relationship between Pick's Theorem and the positivity criterion for analyticity. Yemon Choi's answer points out a better way to think about the latter; it is a simple criterion for being a multiplier of a reproducing kernel Hilbert space, and it works even for those RKHS for which the analogue of Pick's Theorem is false. The general criterion appears in section 2.3 of Agler and McCarthy's book *Pick Interpolation and Hilbert Function Spaces*, the identification of the multipliers of Hardy space appears in section 3.4, and Pick's Theorem is proved in section 5.2.