Interpolation by polynomials has been studied for a long time, being probably formalized for the first time by Newton at the end of the 17th century and formally proved by Lagrange in the 18th century. In this setting one looks for a polynomial taking given values in a finite number of points. Naturally, one could ask whether it is possible to interpolate in infinitely many points. This leads to two new questions: by which functions do we wish to interpolate, and which values do we expect to be interpolable? The interpolating functions should be somehow rigid, and thus analytic functions seem to be a good first guess. The simplest case for the data we would like to interpolate is just the constant value 0. Obviously, the constant (entire) function 0 interpolates these values, but, less trivially, the Weierstrass factorization theorem yields a non trivial solution in this setting leading us somewhere to the mid-19th century. It is clear that for this non trivial solution to exist, the points in which we want to interpolate cannot accumulate in a finite point of the complex plane. The Mittag-Leffler theorem (published in Acta Math in 1884) then allows to interpolate arbitrary values on such discrete sequences, and even derivatives to arbitrary finite orders (initially this theorem considers meromorphic functions). If we add more structure to the set of holomorphic functions, e.g. being in a Frechet-, Banach-,or Hilbert-space of analytic functions (on a domain, e.g. the unit disk, a half plane, the entire complex plane, etc.), we can no longer expect arbitrary values to be interpolable, and structure has to be added also to the value sequences (for instance being in a suitable sequence space).

Sampling problems are closely related with interpolation problems. Sampling means that for a given function in a space of holomorphic functions, it can be recovered from partial information in a stable way. By this we mean that the partial information allows not only to recover the function uniquely but also to estimate its norm (we will only consider normed spaces, though more general settings can be discussed). If the partial information is given in points, recovering the function from these points is a kind of interpolation problem (but possible redundance will not allow to interpolate all the data from an a priori given sequence space as mentioned above in the general context of interpolation problems).

A special situation occurs when the points in which we interpolate define an interpolating and sampling sequence at the same time. Such sequencess are sometimes called complete interpolating sequences. The most prominent example in this setting is probably the Shannon-Kotelnikov-Whittaker sampling theorem (1949), which states that functions from the Paley-Wiener space $PW_{\pi}$ can be recovered from their values on the integers in a stable way.

This last remark points at the wealth of applications of interpolation and sampling problems. Indeed, the above mentioned sampling theorem is a central ingredient in signal theory and at the base of modern digital audio devices, but these problems have many more applications for instance in function theory itself, in operator theory, in control theory, etc.

In this course we will mainly focus on interpolation problems, pointing occasionally to the sampling setting. The first aim will be to discuss the above mentioned interplay between the class of functions we use for the interpolation and the values we want to interpolate, the so-called trace. This will be done for two special cases for which the trace is quite intuitive: uniformly bounded analytic function and reproducing kernel Hilbert spaces. After two motivating examples,and some general observations (linking the problem with unconditional or Riesz sequences and bases), we will present the main steps for the interpolation theorem in the setting of the Hardy space (mostly due to Carleson in the early 60's).

Interpolation problems, even in the infinite setting, are often solved in the spirit of Lagrange: construct functions vanishing in all interpolation points but one where it takes the value 1 (or a characteristic value given by the characteristic growth of functions in the space under consideration) as building blocks of an interpolation formula. Another road is to construct first a smooth interpolating function $F$ (which is easy!), and then correct it to obtain the required holomorphic interpolating function (which is the difficult part). Clearly, the part we add (or subtract) to the smooth interpolating function mustn't alter the interpolated values and it has to kill the $\overline{\partial}$-part of $F$, leading thus to an application of Hormander $\overline{\partial}$-techniques. In this setting, another aim of this course is to present the general method of Berndtsson-Ortega-Cerda (1995) for the construction of the interpolating function in the special case of the Fock space for which Seip and Wallsten gave a characterization in 1992.

We finish the presentation, if time allows, with an overview of different possible generalizations: generalized interpolation (e.g. multiple interpolation), random interpolation, constrained interpolation.

Below is a short list of some relevant literature for the lecture. There are also books by Eric Sawyer (Function theory: interpolation and corona problems, published by the Fields Institute in 2009), and by Jim Agler and John McCarthy (Pick interpolation and Hilbert function spaces, 2004), which are not explicitly mentioned in the presentation.

References:

A. Aleman, M. Hartz, J. McCarthy, S. Richter}, Interpolating Sequences in Spaces with the Complete Pick Property, Int.

Math. Res. Not. 2019, no. 12, 3832-3854.

B. Berndtsson, J.~Ortega-Cerda, On interpolation and sampling in Hilbert spaces of analytic functions}, J. Reine Angew. Math. 464 (1995), 109-128.

Garnett, John B.Bounded analytic functions. Revised first edition. Graduate Texts in Mathematics, 236. Springer, New York, 2007. xiv+459 pp.

N, Nikolski, Operators, Functions, and Systems: an Easy Reading. Vol. 2, Math.Surveys Monogr., 93, AMS, Providence, RI, 2002.

K, Seip, Density theorems for sampling and interpolation in the Bargmann--Fock space I, J. Reine Angew.Math. 429 (1992) 91-106.

K, Seip, Interpolating and Sampling in Spaces of Analytic Functions, American Mathematical Society, Providence, 2004.

K, Seip, R.Wallsten, Density theorems for sampling and interpolation in the Bargmann--Fock space II, J. Reine Angew. Math.429 (1992) 107-113.