I understand that Sarason generalized the interpolation problem by taking it into the operator theoretic setting via reproducing kernels, but whose idea was it to use reproducing kernels such as the Szegö kernel in the first place?


I've had a look at Sarasons -67 paper and it doesn't seem that it was his idea.



Who was first to use reproducing kernels in order to try to solve interpolation problems?
You might look at Carleson's 1958 paper "An interpolation problem for bounded analytic functions". A modern treatment is given in Agler and McCarthy's *Pick Interpolation and Hilbert Function Spaces*, Chapter 9.


